Theorem fneq12 5454

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 5453	1 |- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   Fn wfn 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-fun 5359  df-fn 5360

This theorem is referenced by:  tfrlem3ag  6553  tfrlem3a  6554  tfr1onlem3ag  6581  frecfnom  6645  xnn0nnen  10823