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Theorem mpt2exxg2 41425
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 mpt2exxg 7196. (Contributed by AV, 30-Mar-2019.)
Hypothesis
Ref Expression
mpt2exxg2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
mpt2exxg2 ((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 mpt2exxg2
StepHypRef Expression
1 mpt2exxg2.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21mpt2fun 6722 . 2 Fun 𝐹
31dmmpt2ssx2 41424 . . 3 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
4 snex 4874 . . . . . 6 {𝑦} ∈ V
5 xpexg 6920 . . . . . 6 ((𝐴𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
64, 5mpan2 706 . . . . 5 (𝐴𝑆 → (𝐴 × {𝑦}) ∈ V)
76ralimi 2947 . . . 4 (∀𝑦𝐵 𝐴𝑆 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
8 iunexg 7096 . . . 4 ((𝐵𝑅 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
97, 8sylan2 491 . . 3 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 ssexg 4769 . . 3 ((dom 𝐹 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → dom 𝐹 ∈ V)
113, 9, 10sylancr 694 . 2 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → dom 𝐹 ∈ V)
12 funex 6442 . 2 ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V)
132, 11, 12sylancr 694 1 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  wss 3559  {csn 4153   ciun 4490   × cxp 5077  dom cdm 5079  Fun wfun 5846  cmpt2 6612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121
This theorem is referenced by:  lincop  41506
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