Theorem f1odm 5290

Description: The domain of a one-to-one onto mapping. (Contributed by set.mm contributors, 8-Mar-2014.)

Assertion

Ref	Expression
f1odm	⊢ (F:A–1-1-onto→B → dom F = A)

Proof of Theorem f1odm

Step	Hyp	Ref	Expression
1		f1ofn 5288	. 2 ⊢ (F:A–1-1-onto→B → F Fn A)
2		fndm 5182	. 2 ⊢ (F Fn A → dom F = A)
3	1, 2	syl 15	1 ⊢ (F:A–1-1-onto→B → dom F = A)

Syntax hints:   → wi 4   = wceq 1642  dom cdm 4772   Fn wfn 4776  –1-1-onto→wf1o 4780

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-fn 4790  df-f 4791  df-f1 4792  df-f1o 4794

This theorem is referenced by:  bren  6030  enpw1  6062  enmap1lem5  6073  nenpw1pwlem2  6085  ncdisjun  6136  sbthlem3  6205