Theorem fneq12 5224

Description: Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Assertion

Ref	Expression
fneq12	|- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) )

Proof of Theorem fneq12

Step	Hyp	Ref	Expression
1		simpl 108	. 2 |- ( ( F = G /\ A = B ) -> F = G )
2		simpr 109	. 2 |- ( ( F = G /\ A = B ) -> A = B )
3	1, 2	fneq12d 5223	1 |- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   Fn wfn 5126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-fun 5133  df-fn 5134

This theorem is referenced by:  tfrlem3ag  6214  tfrlem3a  6215  tfr1onlem3ag  6242  frecfnom  6306