Theorem mptexgf 7259

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 6616	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		eqid 2740	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	dmmpt 6271	. . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
4		tru 1541	. . . . . . 7 ⊢ ⊤
5	4	2a1i 12	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ V → ⊤))
6	5	ss2rabi 4100	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
7		mptexgf.a	. . . . . 6 ⊢ Ⅎ𝑥𝐴
8	7	rabtru 3705	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
9	6, 8	sseqtri 4045	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
10	3, 9	eqsstri 4043	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
11		ssexg 5341	. . 3 ⊢ ((dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
12	10, 11	mpan 689	. 2 ⊢ (𝐴 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
13		funex 7256	. 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
14	1, 12, 13	sylancr 586	1 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)

Syntax hints:   → wi 4  ⊤wtru 1538   ∈ wcel 2108  Ⅎwnfc 2893  {crab 3443  Vcvv 3488   ⊆ wss 3976   ↦ cmpt 5249  dom cdm 5700  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  esumrnmpt2  34032  exrecfnlem  37345  mptexf  45145