Theorem nffn 6452

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 6358	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		nffn.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3	2	nffun 6378	. . 3 ⊢ Ⅎ𝑥Fun 𝐹
4	2	nfdm 5823	. . . 4 ⊢ Ⅎ𝑥dom 𝐹
5		nffn.2	. . . 4 ⊢ Ⅎ𝑥𝐴
6	4, 5	nfeq 2991	. . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴
7	3, 6	nfan 1900	. 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
8	1, 7	nfxfr 1853	1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴

Syntax hints:   ∧ wa 398   = wceq 1537  Ⅎwnf 1784  Ⅎwnfc 2961  dom cdm 5555  Fun wfun 6349   Fn wfn 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-fun 6357  df-fn 6358

This theorem is referenced by:  nff  6510  nffo  6589  feqmptdf  6735  nfixpw  8480  nfixp  8481  nfixp1  8482  bnj1463  32327  choicefi  41483  stoweidlem31  42336  stoweidlem35  42340  stoweidlem59  42364