Theorem mptexgf 7158

Description: If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) (Revised by Thierry Arnoux, 17-May-2020.)

Hypothesis

Ref	Expression
mptexgf.a	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
mptexgf	⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)

Proof of Theorem mptexgf

Step	Hyp	Ref	Expression
1		funmpt 6520	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		eqid 2729	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	dmmpt 6189	. . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
4		tru 1544	. . . . . . 7 ⊢ ⊤
5	4	2a1i 12	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ V → ⊤))
6	5	ss2rabi 4028	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
7		mptexgf.a	. . . . . 6 ⊢ Ⅎ𝑥𝐴
8	7	rabtru 3645	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
9	6, 8	sseqtri 3984	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
10	3, 9	eqsstri 3982	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
11		ssexg 5262	. . 3 ⊢ ((dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
12	10, 11	mpan 690	. 2 ⊢ (𝐴 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
13		funex 7155	. 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
14	1, 12, 13	sylancr 587	1 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)

Syntax hints:   → wi 4  ⊤wtru 1541   ∈ wcel 2109  Ⅎwnfc 2876  {crab 3394  Vcvv 3436   ⊆ wss 3903   ↦ cmpt 5173  dom cdm 5619  Fun wfun 6476

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-rep 5218  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-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  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-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490

This theorem is referenced by:  esumrnmpt2  34035  exrecfnlem  37353  mptexf  45215