Theorem f1oeq3d 5477

Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
f1oeq3d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
f1oeq3d	|- ( ph -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )

Proof of Theorem f1oeq3d

Step	Hyp	Ref	Expression
1		f1oeq3d.1	. 2 |- ( ph -> A = B )
2		f1oeq3 5470	. 2 |- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )
3	1, 2	syl 14	1 |- ( ph -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   -1-1-onto->wf1o 5234

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242

This theorem is referenced by:  fprodssdc  11630  fprodcnv  11665