Theorem nffn 6596

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 6494	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		nffn.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3	2	nffun 6519	. . 3 ⊢ Ⅎ𝑥Fun 𝐹
4	2	nfdm 5902	. . . 4 ⊢ Ⅎ𝑥dom 𝐹
5		nffn.2	. . . 4 ⊢ Ⅎ𝑥𝐴
6	4, 5	nfeq 2918	. . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴
7	3, 6	nfan 1902	. 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
8	1, 7	nfxfr 1855	1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴

Syntax hints:   ∧ wa 397   = wceq 1541  Ⅎwnf 1785  Ⅎwnfc 2885  dom cdm 5630  Fun wfun 6485   Fn wfn 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-fun 6493  df-fn 6494

This theorem is referenced by:  nff  6659  nffo  6750  feqmptdf  6907  nfixpw  8787  nfixp  8788  nfixp1  8789  bnj1463  33401  choicefi  43123  stoweidlem31  43964  stoweidlem35  43968  stoweidlem59  43992