Theorem nffn 5350

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 5257	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		nffn.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3	2	nffun 5277	. . 3 ⊢ Ⅎ𝑥Fun 𝐹
4	2	nfdm 4906	. . . 4 ⊢ Ⅎ𝑥dom 𝐹
5		nffn.2	. . . 4 ⊢ Ⅎ𝑥𝐴
6	4, 5	nfeq 2344	. . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴
7	3, 6	nfan 1576	. 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
8	1, 7	nfxfr 1485	1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴

Syntax hints:   ∧ wa 104   = wceq 1364  Ⅎwnf 1471  Ⅎwnfc 2323  dom cdm 4659  Fun wfun 5248   Fn wfn 5249

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-fun 5256  df-fn 5257

This theorem is referenced by:  nff  5400  nffo  5475  nfixpxy  6771  nfixp1  6772