Theorem mpoexxg2 48299

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 8033. (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 7493	. 2 ⊢ Fun 𝐹
3	1	dmmpossx2 48298	. . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
4		vsnex 5384	. . . . . 6 ⊢ {𝑦} ∈ V
5		xpexg 7706	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
6	4, 5	mpan2 691	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐴 × {𝑦}) ∈ V)
7	6	ralimi 3066	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
8		iunexg 7921	. . . 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 5273	. . 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 7175	. 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 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ⊆ wss 3911  {csn 4585  ∪ ciun 4951   × cxp 5629  dom cdm 5631  Fun wfun 6493   ∈ cmpo 7371

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948

This theorem is referenced by:  lincop  48370