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Theorem mpoexxg2 47401
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 8080. (Contributed by AV, 30-Mar-2019.)
Hypothesis
Ref Expression
mpoexxg2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
mpoexxg2 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 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
StepHypRef Expression
1 mpoexxg2.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21mpofun 7544 . 2 Fun 𝐹
31dmmpossx2 47400 . . 3 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
4 vsnex 5431 . . . . . 6 {𝑦} ∈ V
5 xpexg 7752 . . . . . 6 ((𝐴𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
64, 5mpan2 690 . . . . 5 (𝐴𝑆 → (𝐴 × {𝑦}) ∈ V)
76ralimi 3080 . . . 4 (∀𝑦𝐵 𝐴𝑆 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
8 iunexg 7967 . . . 4 ((𝐵𝑅 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
97, 8sylan2 592 . . 3 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 ssexg 5323 . . 3 ((dom 𝐹 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → dom 𝐹 ∈ V)
113, 9, 10sylancr 586 . 2 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → dom 𝐹 ∈ V)
12 funex 7231 . 2 ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V)
132, 11, 12sylancr 586 1 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3058  Vcvv 3471  wss 3947  {csn 4629   ciun 4996   × cxp 5676  dom cdm 5678  Fun wfun 6542  cmpo 7422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994
This theorem is referenced by:  lincop  47476
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