Theorem mpoexxg2 45561

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 7889. (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 7376	. 2 ⊢ Fun 𝐹
3	1	dmmpossx2 45560	. . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
4		snex 5349	. . . . . 6 ⊢ {𝑦} ∈ V
5		xpexg 7578	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
6	4, 5	mpan2 687	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐴 × {𝑦}) ∈ V)
7	6	ralimi 3086	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
8		iunexg 7779	. . . 4 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
9	7, 8	sylan2 592	. . 3 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
10		ssexg 5242	. . 3 ⊢ ((dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → dom 𝐹 ∈ V)
11	3, 9, 10	sylancr 586	. 2 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → dom 𝐹 ∈ V)
12		funex 7077	. 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V)
13	2, 11, 12	sylancr 586	1 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  {csn 4558  ∪ ciun 4921   × cxp 5578  dom cdm 5580  Fun wfun 6412   ∈ cmpo 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805

This theorem is referenced by:  lincop  45637