Theorem mpoexxg2 47003

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 8061. (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 7531	. 2 ⊢ Fun 𝐹
3	1	dmmpossx2 47002	. . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
4		vsnex 5429	. . . . . 6 ⊢ {𝑦} ∈ V
5		xpexg 7736	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
6	4, 5	mpan2 689	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐴 × {𝑦}) ∈ V)
7	6	ralimi 3083	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
8		iunexg 7949	. . . 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 5323	. . 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 7220	. 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ⊆ wss 3948  {csn 4628  ∪ ciun 4997   × cxp 5674  dom cdm 5676  Fun wfun 6537   ∈ cmpo 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975

This theorem is referenced by:  lincop  47079