Theorem f1oeq2 5118

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

Assertion

Ref	Expression
f1oeq2	|- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )

Proof of Theorem f1oeq2

Step	Hyp	Ref	Expression
1		f1eq2 5088	. . 3 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )
2		foeq2 5103	. . 3 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )
3	1, 2	anbi12d 442	. 2 |- ( A = B -> ( ( F : A -1-1-> C /\ F : A -onto-> C ) <-> ( F : B -1-1-> C /\ F : B -onto-> C ) ) )
4		df-f1o 4909	. 2 |- ( F : A -1-1-onto-> C <-> ( F : A -1-1-> C /\ F : A -onto-> C ) )
5		df-f1o 4909	. 2 |- ( F : B -1-1-onto-> C <-> ( F : B -1-1-> C /\ F : B -onto-> C ) )
6	3, 4, 5	3bitr4g 212	1 |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   -1-1->wf1 4899   -onto->wfo 4900   -1-1-onto->wf1o 4901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909

This theorem is referenced by:  f1oeq23  5120  f1oeq123d  5123  f1osng  5167  isoeq4  5444  bren  6228