Theorem f1eq2 5597

Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Assertion

Ref	Expression
f1eq2	|- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )

Proof of Theorem f1eq2

Step	Hyp	Ref	Expression
1		feq2 5520	. . 3 |- ( A = B -> ( F : A --> C <-> F : B --> C ) )
2	1	anbi1d 469	. 2 |- ( A = B -> ( ( F : A --> C /\ Fun `' F ) <-> ( F : B --> C /\ Fun `' F ) ) )
3		df-f1 5385	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
4		df-f1 5385	. 2 |- ( F : B -1-1-> C <-> ( F : B --> C /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   `'ccnv 4775   Fun wfun 5374   -->wf 5376   -1-1->wf1 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-fn 5383  df-f 5384  df-f1 5385

This theorem is used by:  f1oeq2  5631  f1eq123d  5634  f10d  5678  brdom2g  7035  brdomg  7036  dom1o  7120  hashf1  11309  ennnfonelemen  13370  ausgrusgrben  16614  usgr0  16685  uspgr1edc  16686