Theorem fnmptf 6618

Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)

Hypothesis

Ref	Expression
mptfnf.0	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
fnmptf	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

Proof of Theorem fnmptf

Step	Hyp	Ref	Expression
1		elex 3457	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2	1	ralimi 3066	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
3		mptfnf.0	. . 3 ⊢ Ⅎ𝑥𝐴
4	3	mptfnf 6617	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
5	2, 4	sylib 218	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109  Ⅎwnfc 2876  ∀wral 3044  Vcvv 3436   ↦ cmpt 5173   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  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  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-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-fun 6484  df-fn 6485

This theorem is referenced by:  offval2f  7628  esumgsum  34018  esumc  34024  bj-mptval  37101  aks4d1p1p5  42058  rfovcnvf1od  43987  dssmapf1od  44004  ntrrn  44105  dssmapntrcls  44111