Theorem f1oeq2 5493

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 5459	. . 3 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )
2		foeq2 5477	. . 3 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )
3	1, 2	anbi12d 473	. 2 |- ( A = B -> ( ( F : A -1-1-> C /\ F : A -onto-> C ) <-> ( F : B -1-1-> C /\ F : B -onto-> C ) ) )
4		df-f1o 5265	. 2 |- ( F : A -1-1-onto-> C <-> ( F : A -1-1-> C /\ F : A -onto-> C ) )
5		df-f1o 5265	. 2 |- ( F : B -1-1-onto-> C <-> ( F : B -1-1-> C /\ F : B -onto-> C ) )
6	3, 4, 5	3bitr4g 223	1 |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   -1-1->wf1 5255   -onto->wfo 5256   -1-1-onto->wf1o 5257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265

This theorem is referenced by:  f1oeq23  5495  f1oeq123d  5498  f1oeq2d  5500  f1osng  5545  isoeq4  5851  bren  6806  f1dmvrnfibi  7010  summodclem3  11545  summodclem2a  11546  summodc  11548  fsum3  11552  fsumf1o  11555  sumsnf  11574  fprodf1o  11753  prodsnf  11757  znfi  14211  znhash  14212