verb/mɛ.tʁə sa mɛ̃ o fø/
to bet one's boots that, to bet one's bottom dollar that
“Je mettrais ma main au feu qu'ils ont baisé.” — I would put my hand in the fire that they fucked; I would bet my bottom dollar that they fucked.
Synonyms: donner sa main à couper, mettre sa main à couper, mettre sa tête à couper
Literally, “to put one's hand in fire”. Compare German für etwas die Hand ins Feuer legen.