Theorem mpoexxg2 48259

Description: Existence of an operation class abstraction (version for dependent domains, i.e. the first base class may depend on the second base class), analogous to mpoexxg 8101. (Contributed by AV, 30-Mar-2019.)

Hypothesis

Ref	Expression
mpoexxg2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
mpoexxg2	⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → 𝐹 ∈ V)

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

Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpoexxg2

Step	Hyp	Ref	Expression
1		mpoexxg2.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	mpofun 7558	. 2 ⊢ Fun 𝐹
3	1	dmmpossx2 48258	. . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
4		vsnex 5433	. . . . . 6 ⊢ {𝑦} ∈ V
5		xpexg 7771	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
6	4, 5	mpan2 691	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐴 × {𝑦}) ∈ V)
7	6	ralimi 3082	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
8		iunexg 7989	. . . 4 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
9	7, 8	sylan2 593	. . 3 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
10		ssexg 5322	. . 3 ⊢ ((dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → dom 𝐹 ∈ V)
11	3, 9, 10	sylancr 587	. 2 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → dom 𝐹 ∈ V)
12		funex 7240	. 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V)
13	2, 11, 12	sylancr 587	1 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ⊆ wss 3950  {csn 4625  ∪ ciun 4990   × cxp 5682  dom cdm 5684  Fun wfun 6554   ∈ cmpo 7434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016

This theorem is referenced by:  lincop  48330