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Theorem eliunxp 5848
Description: Membership in a union of Cartesian products. Analogue of elxp 5708 for nonconstant 𝐵(𝑥). (Contributed by Mario Carneiro, 29-Dec-2014.)
Assertion
Ref Expression
eliunxp (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem eliunxp
StepHypRef Expression
1 relxp 5703 . . . . . 6 Rel ({𝑥} × 𝐵)
21rgenw 3065 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
3 reliun 5826 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
42, 3mpbir 231 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
5 elrel 5808 . . . 4 ((Rel 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 690 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
76pm4.71ri 560 . 2 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
8 nfiu1 5027 . . . 4 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
98nfel2 2924 . . 3 𝑥 𝐶 𝑥𝐴 ({𝑥} × 𝐵)
10919.41 2235 . 2 (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
11 19.41v 1949 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
12 eleq1 2829 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
13 opeliunxp 5752 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
1412, 13bitrdi 287 . . . . . 6 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
1514pm5.32i 574 . . . . 5 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1615exbii 1848 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1711, 16bitr3i 277 . . 3 ((∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1817exbii 1848 . 2 (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
197, 10, 183bitr2i 299 1 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  {csn 4626  cop 4632   ciun 4991   × cxp 5683  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  raliunxp  5850  dfmpt3  6702  mpomptx  7546  fsumcom2  15810  fprodcom2  16020  isfunc  17909  gsum2d2  19992  dprd2d2  20064  fsumvma  27257  2ndresdju  32659  mpomptxf  32687  gsumwrd2dccatlem  33069  poimirlem26  37653  dvnprodlem1  45961
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