Theorem nffn 5452

Description: Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)

Hypotheses

Ref	Expression
nffn.1	|- F/_ x F
nffn.2	|- F/_ x A

Assertion

Ref	Expression
nffn	|- F/ x F Fn A

Proof of Theorem nffn

Step	Hyp	Ref	Expression
1		df-fn 5355	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2		nffn.1	. . . 4 |- F/_ x F
3	2	nffun 5375	. . 3 |- F/ x Fun F
4	2	nfdm 5001	. . . 4 |- F/_ x dom F
5		nffn.2	. . . 4 |- F/_ x A
6	4, 5	nfeq 2392	. . 3 |- F/ x dom F = A
7	3, 6	nfan 1614	. 2 |- F/ x ( Fun F /\ dom F = A )
8	1, 7	nfxfr 1523	1 |- F/ x F Fn A

Syntax hints:   /\ wa 104   = wceq 1398   F/wnf 1509   F/_wnfc 2371   dom cdm 4749   Fun wfun 5346   Fn wfn 5347

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-fun 5354  df-fn 5355

This theorem is referenced by:  nff  5505  nffo  5589  nfixpxy  6952  nfixp1  6953