Theorem fneq12 5367

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 5366	1 |- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   Fn wfn 5266

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-fun 5273  df-fn 5274

This theorem is referenced by:  tfrlem3ag  6395  tfrlem3a  6396  tfr1onlem3ag  6423  frecfnom  6487  xnn0nnen  10582