Theorem f1oeq1d 5499

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 5492	. 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 1364   -1-1-onto->wf1o 5257

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265

This theorem is referenced by:  grplactcnv  13234  eqgen  13357