Theorem mpoexxg2 44460

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 7766. (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 7269	. 2 ⊢ Fun 𝐹
3	1	dmmpossx2 44459	. . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
4		snex 5325	. . . . . 6 ⊢ {𝑦} ∈ V
5		xpexg 7466	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
6	4, 5	mpan2 689	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐴 × {𝑦}) ∈ V)
7	6	ralimi 3159	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
8		iunexg 7657	. . . 4 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
9	7, 8	sylan2 594	. . 3 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
10		ssexg 5220	. . 3 ⊢ ((dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → dom 𝐹 ∈ V)
11	3, 9, 10	sylancr 589	. 2 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → dom 𝐹 ∈ V)
12		funex 6975	. 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V)
13	2, 11, 12	sylancr 589	1 ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3137  Vcvv 3491   ⊆ wss 3929  {csn 4560  ∪ ciun 4912   × cxp 5546  dom cdm 5548  Fun wfun 6342   ∈ cmpo 7151

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-oprab 7153  df-mpo 7154  df-1st 7682  df-2nd 7683

This theorem is referenced by:  lincop  44537