Theorem abrexexd 32495

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 5939	. . 3 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2		df-mpt 5207	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	2	rneqi 5922	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4		df-rex 3062	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
5	4	abbii 2803	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	1, 3, 5	3eqtr4i 2769	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
7		abrexexd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
8		funmpt 6579	. . . 4 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
9		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	9	dmmpt 6234	. . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
11		abrexexd.0	. . . . . 6 ⊢ Ⅎ𝑥𝐴
12	11	rabexgfGS 32485	. . . . 5 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ∈ V)
13	10, 12	eqeltrid 2839	. . . 4 ⊢ (𝐴 ∈ V → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
14		funex 7216	. . . 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 7903	. . 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 2840	1 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2714  Ⅎwnfc 2884  ∃wrex 3061  {crab 3420  Vcvv 3464  {copab 5186   ↦ cmpt 5206  dom cdm 5659  ran crn 5660  Fun wfun 6530

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  esumc  34087