Theorem f1oeq1d 5632

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 5625	. 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 1402   -1-1-onto->wf1o 5374

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382

This theorem is referenced by:  grplactcnv  13887  eqgen  14010  domomsubct  16948