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Theorem mpt2exxg2 42618
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 7404. (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 6919 . 2 Fun 𝐹
31dmmpt2ssx2 42617 . . 3 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
4 snex 5049 . . . . . 6 {𝑦} ∈ V
5 xpexg 7117 . . . . . 6 ((𝐴𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
64, 5mpan2 709 . . . . 5 (𝐴𝑆 → (𝐴 × {𝑦}) ∈ V)
76ralimi 3082 . . . 4 (∀𝑦𝐵 𝐴𝑆 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
8 iunexg 7300 . . . 4 ((𝐵𝑅 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
97, 8sylan2 492 . . 3 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 ssexg 4948 . . 3 ((dom 𝐹 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → dom 𝐹 ∈ V)
113, 9, 10sylancr 698 . 2 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → dom 𝐹 ∈ V)
12 funex 6638 . 2 ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V)
132, 11, 12sylancr 698 1 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  wss 3707  {csn 4313   ciun 4664   × cxp 5256  dom cdm 5258  Fun wfun 6035  cmpt2 6807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326
This theorem is referenced by:  lincop  42699
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