Theorem mptexgf 6628

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 6069	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		eqid 2770	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	dmmpt 5774	. . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
4		a1tru 1647	. . . . . . 7 ⊢ (𝐵 ∈ V → ⊤)
5	4	rgenw 3072	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ∈ V → ⊤)
6		ss2rab 3825	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ ∀𝑥 ∈ 𝐴 (𝐵 ∈ V → ⊤))
7	5, 6	mpbir 221	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
8		mptexgf.a	. . . . . 6 ⊢ Ⅎ𝑥𝐴
9	8	rabtru 3510	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
10	7, 9	sseqtri 3784	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
11	3, 10	eqsstri 3782	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
12		ssexg 4935	. . 3 ⊢ ((dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
13	11, 12	mpan 662	. 2 ⊢ (𝐴 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
14		funex 6625	. 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
15	1, 13, 14	sylancr 567	1 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)

Syntax hints:   → wi 4  ⊤wtru 1631   ∈ wcel 2144  Ⅎwnfc 2899  ∀wral 3060  {crab 3064  Vcvv 3349   ⊆ wss 3721   ↦ cmpt 4861  dom cdm 5249  Fun wfun 6025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039

This theorem is referenced by:  numclwwlk1lem2  27544  esumrnmpt2  30464  mptexf  39956