Theorem f1oeq1d 5445

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

Hypothesis

Ref	Expression
f1oeq1d.1	|- ( ph -> F = G )

Assertion

Ref	Expression
f1oeq1d	|- ( ph -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )

Proof of Theorem f1oeq1d

Step	Hyp	Ref	Expression
1		f1oeq1d.1	. 2 |- ( ph -> F = G )
2		f1oeq1 5438	. 2 |- ( F = G -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )
3	1, 2	syl 14	1 |- ( ph -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1351   -1-1-onto->wf1o 5204

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-io 707  ax-5 1443  ax-7 1444  ax-gen 1445  ax-ie1 1489  ax-ie2 1490  ax-8 1500  ax-10 1501  ax-11 1502  ax-i12 1503  ax-bndl 1505  ax-4 1506  ax-17 1522  ax-i9 1526  ax-ial 1530  ax-i5r 1531  ax-ext 2155

This theorem depends on definitions:  df-bi 117  df-3an 978  df-tru 1354  df-nf 1457  df-sb 1759  df-clab 2160  df-cleq 2166  df-clel 2169  df-nfc 2304  df-v 2735  df-un 3128  df-in 3130  df-ss 3137  df-sn 3592  df-pr 3593  df-op 3595  df-br 3996  df-opab 4057  df-rel 4624  df-cnv 4625  df-co 4626  df-dm 4627  df-rn 4628  df-fun 5207  df-fn 5208  df-f 5209  df-f1 5210  df-fo 5211  df-f1o 5212

This theorem is referenced by:  grplactcnv  12828