Theorem fneq12 5122

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 5121	1 |- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   Fn wfn 5025

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-fun 5032  df-fn 5033

This theorem is referenced by:  tfrlem3ag  6090  tfrlem3a  6091  tfr1onlem3ag  6118  frecfnom  6182