Theorem nffn 5314

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 5221	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2		nffn.1	. . . 4 |- F/_ x F
3	2	nffun 5241	. . 3 |- F/ x Fun F
4	2	nfdm 4873	. . . 4 |- F/_ x dom F
5		nffn.2	. . . 4 |- F/_ x A
6	4, 5	nfeq 2327	. . 3 |- F/ x dom F = A
7	3, 6	nfan 1565	. 2 |- F/ x ( Fun F /\ dom F = A )
8	1, 7	nfxfr 1474	1 |- F/ x F Fn A

Syntax hints:   /\ wa 104   = wceq 1353   F/wnf 1460   F/_wnfc 2306   dom cdm 4628   Fun wfun 5212   Fn wfn 5213

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-fun 5220  df-fn 5221

This theorem is referenced by:  nff  5364  nffo  5439  nfixpxy  6719  nfixp1  6720