Theorem fneq12 5265

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 5264	1 |- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   Fn wfn 5167

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-fun 5174  df-fn 5175

This theorem is referenced by:  tfrlem3ag  6258  tfrlem3a  6259  tfr1onlem3ag  6286  frecfnom  6350