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Theorem mpoexxg2 46503
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 8012. (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 7484 . 2 Fun 𝐹
31dmmpossx2 46502 . . 3 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
4 vsnex 5390 . . . . . 6 {𝑦} ∈ V
5 xpexg 7688 . . . . . 6 ((𝐴𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
64, 5mpan2 690 . . . . 5 (𝐴𝑆 → (𝐴 × {𝑦}) ∈ V)
76ralimi 3083 . . . 4 (∀𝑦𝐵 𝐴𝑆 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
8 iunexg 7900 . . . 4 ((𝐵𝑅 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
97, 8sylan2 594 . . 3 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 ssexg 5284 . . 3 ((dom 𝐹 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → dom 𝐹 ∈ V)
113, 9, 10sylancr 588 . 2 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → dom 𝐹 ∈ V)
12 funex 7173 . 2 ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V)
132, 11, 12sylancr 588 1 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3447  wss 3914  {csn 4590   ciun 4958   × cxp 5635  dom cdm 5637  Fun wfun 6494  cmpo 7363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926
This theorem is referenced by:  lincop  46579
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