Theorem abrexexd 32484

Description: Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)

Hypotheses

Ref	Expression
abrexexd.0	⊢ Ⅎ𝑥𝐴
abrexexd.1	⊢ (𝜑 → 𝐴 ∈ V)

Assertion

Ref	Expression
abrexexd	⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem abrexexd

Step	Hyp	Ref	Expression
1		rnopab 5894	. . 3 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2		df-mpt 5173	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	2	rneqi 5877	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4		df-rex 3057	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
5	4	abbii 2798	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	1, 3, 5	3eqtr4i 2764	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
7		abrexexd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
8		funmpt 6519	. . . 4 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
9		eqid 2731	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	9	dmmpt 6187	. . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
11		abrexexd.0	. . . . . 6 ⊢ Ⅎ𝑥𝐴
12	11	rabexgfGS 32474	. . . . 5 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ∈ V)
13	10, 12	eqeltrid 2835	. . . 4 ⊢ (𝐴 ∈ V → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
14		funex 7153	. . . 4 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
15	8, 13, 14	sylancr 587	. . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
16		rnexg 7832	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
17	7, 15, 16	3syl 18	. 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
18	6, 17	eqeltrrid 2836	1 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  Ⅎwnfc 2879  ∃wrex 3056  {crab 3395  Vcvv 3436  {copab 5153   ↦ cmpt 5172  dom cdm 5616  ran crn 5617  Fun wfun 6475

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489

This theorem is referenced by:  esumc  34059