Theorem f1oeq3d 5580

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 5573	. 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 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-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333

This theorem is referenced by:  fprodssdc  12150  fprodcnv  12185  pwssnf1o  13380  uspgrf1oedg  16026  usgrf1oedg  16055  eupthres  16307  gfsumval  16680