Theorem f1oeq3d 5589

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 5582	. 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 1398   -1-1-onto->wf1o 5332

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340

This theorem is referenced by:  fprodssdc  12231  fprodcnv  12266  pwssnf1o  13461  uspgrf1oedg  16117  usgrf1oedg  16146  eupthres  16398  gfsumval  16809