Theorem f1oeq2d 5579

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

Hypothesis

Ref	Expression
f1oeq2d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
f1oeq2d	|- ( ph -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )

Proof of Theorem f1oeq2d

Step	Hyp	Ref	Expression
1		f1oeq2d.1	. 2 |- ( ph -> A = B )
2		f1oeq2 5572	. 2 |- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
3	1, 2	syl 14	1 |- ( ph -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   -1-1-onto->wf1o 5325

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333

This theorem is referenced by:  prodmodclem3  12135  prodmodc  12138  fprodseq  12143  gsumgfsum1  16681