Theorem nffn 6478

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 6383	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		nffn.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3	2	nffun 6403	. . 3 ⊢ Ⅎ𝑥Fun 𝐹
4	2	nfdm 5820	. . . 4 ⊢ Ⅎ𝑥dom 𝐹
5		nffn.2	. . . 4 ⊢ Ⅎ𝑥𝐴
6	4, 5	nfeq 2917	. . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴
7	3, 6	nfan 1907	. 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
8	1, 7	nfxfr 1860	1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴

Syntax hints:   ∧ wa 399   = wceq 1543  Ⅎwnf 1791  Ⅎwnfc 2884  dom cdm 5551  Fun wfun 6374   Fn wfn 6375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-fun 6382  df-fn 6383

This theorem is referenced by:  nff  6541  nffo  6632  feqmptdf  6782  nfixpw  8597  nfixp  8598  nfixp1  8599  bnj1463  32748  choicefi  42413  stoweidlem31  43247  stoweidlem35  43251  stoweidlem59  43275