Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mpoexxg2 Structured version   Visualization version   GIF version

Theorem mpoexxg2 48259
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 8101. (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 7558 . 2 Fun 𝐹
31dmmpossx2 48258 . . 3 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
4 vsnex 5433 . . . . . 6 {𝑦} ∈ V
5 xpexg 7771 . . . . . 6 ((𝐴𝑆 ∧ {𝑦} ∈ V) → (𝐴 × {𝑦}) ∈ V)
64, 5mpan2 691 . . . . 5 (𝐴𝑆 → (𝐴 × {𝑦}) ∈ V)
76ralimi 3082 . . . 4 (∀𝑦𝐵 𝐴𝑆 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
8 iunexg 7989 . . . 4 ((𝐵𝑅 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
97, 8sylan2 593 . . 3 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 ssexg 5322 . . 3 ((dom 𝐹 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → dom 𝐹 ∈ V)
113, 9, 10sylancr 587 . 2 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → dom 𝐹 ∈ V)
12 funex 7240 . 2 ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V)
132, 11, 12sylancr 587 1 ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  wss 3950  {csn 4625   ciun 4990   × cxp 5682  dom cdm 5684  Fun wfun 6554  cmpo 7434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016
This theorem is referenced by:  lincop  48330
  Copyright terms: Public domain W3C validator