Theorem nffn 6581

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

Hypotheses

Ref	Expression
nffn.1	⊢ Ⅎ𝑥𝐹
nffn.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nffn	⊢ Ⅎ𝑥 𝐹 Fn 𝐴

Proof of Theorem nffn

Step	Hyp	Ref	Expression
1		df-fn 6485	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		nffn.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3	2	nffun 6505	. . 3 ⊢ Ⅎ𝑥Fun 𝐹
4	2	nfdm 5893	. . . 4 ⊢ Ⅎ𝑥dom 𝐹
5		nffn.2	. . . 4 ⊢ Ⅎ𝑥𝐴
6	4, 5	nfeq 2905	. . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴
7	3, 6	nfan 1899	. 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
8	1, 7	nfxfr 1853	1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴

Syntax hints:   ∧ wa 395   = wceq 1540  Ⅎwnf 1783  Ⅎwnfc 2876  dom cdm 5619  Fun wfun 6476   Fn wfn 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-fun 6484  df-fn 6485

This theorem is referenced by:  nff  6648  nffo  6735  feqmptdf  6893  nfixpw  8843  nfixp  8844  nfixp1  8845  bnj1463  35022  choicefi  45178  stoweidlem31  46012  stoweidlem35  46016  stoweidlem59  46040