Theorem fneq12 5414

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 109	. 2 |- ( ( F = G /\ A = B ) -> F = G )
2		simpr 110	. 2 |- ( ( F = G /\ A = B ) -> A = B )
3	1, 2	fneq12d 5413	1 |- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   Fn wfn 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-fun 5320  df-fn 5321

This theorem is referenced by:  tfrlem3ag  6455  tfrlem3a  6456  tfr1onlem3ag  6483  frecfnom  6547  xnn0nnen  10659