Theorem f1oeq3d 5569

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 5562	. 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 1395   -1-1-onto->wf1o 5317

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325

This theorem is referenced by:  fprodssdc  12101  fprodcnv  12136  pwssnf1o  13331  uspgrf1oedg  15974  usgrf1oedg  16003