Theorem nffn 5284

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 5191	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2		nffn.1	. . . 4 |- F/_ x F
3	2	nffun 5211	. . 3 |- F/ x Fun F
4	2	nfdm 4848	. . . 4 |- F/_ x dom F
5		nffn.2	. . . 4 |- F/_ x A
6	4, 5	nfeq 2316	. . 3 |- F/ x dom F = A
7	3, 6	nfan 1553	. 2 |- F/ x ( Fun F /\ dom F = A )
8	1, 7	nfxfr 1462	1 |- F/ x F Fn A

Syntax hints:   /\ wa 103   = wceq 1343   F/wnf 1448   F/_wnfc 2295   dom cdm 4604   Fun wfun 5182   Fn wfn 5183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-fun 5190  df-fn 5191

This theorem is referenced by:  nff  5334  nffo  5409  nfixpxy  6683  nfixp1  6684