Theorem nffn 5417

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 5321	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2		nffn.1	. . . 4 |- F/_ x F
3	2	nffun 5341	. . 3 |- F/ x Fun F
4	2	nfdm 4968	. . . 4 |- F/_ x dom F
5		nffn.2	. . . 4 |- F/_ x A
6	4, 5	nfeq 2380	. . 3 |- F/ x dom F = A
7	3, 6	nfan 1611	. 2 |- F/ x ( Fun F /\ dom F = A )
8	1, 7	nfxfr 1520	1 |- F/ x F Fn A

Syntax hints:   /\ wa 104   = wceq 1395   F/wnf 1506   F/_wnfc 2359   dom cdm 4719   Fun wfun 5312   Fn wfn 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-fun 5320  df-fn 5321

This theorem is referenced by:  nff  5470  nffo  5547  nfixpxy  6864  nfixp1  6865