Theorem f1oeq1d 5495

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 5488	. 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 5253

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261

This theorem is referenced by:  grplactcnv  13174  eqgen  13297