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Theorem eliunxp 5594
Description: Membership in a union of Cartesian products. Analogue of elxp 5466 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 5461 . . . . . 6 Rel ({𝑥} × 𝐵)
21rgenw 3117 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
3 reliun 5575 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
42, 3mpbir 232 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
5 elrel 5557 . . . 4 ((Rel 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 686 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
76pm4.71ri 561 . 2 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
8 nfiu1 4856 . . . 4 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
98nfel2 2965 . . 3 𝑥 𝐶 𝑥𝐴 ({𝑥} × 𝐵)
10919.41 2202 . 2 (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
11 19.41v 1927 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
12 eleq1 2870 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
13 opeliunxp 5505 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
1412, 13syl6bb 288 . . . . . 6 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
1514pm5.32i 575 . . . . 5 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1615exbii 1829 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1711, 16bitr3i 278 . . 3 ((∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1817exbii 1829 . 2 (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
197, 10, 183bitr2i 300 1 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1522  wex 1761  wcel 2081  wral 3105  {csn 4472  cop 4478   ciun 4825   × cxp 5441  Rel wrel 5448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-iun 4827  df-opab 5025  df-xp 5449  df-rel 5450
This theorem is referenced by:  raliunxp  5596  dfmpt3  6350  mpomptx  7121  fsumcom2  14962  fprodcom2  15171  isfunc  16963  gsum2d2  18814  dprd2d2  18883  fsumvma  25471  mpomptxf  30115  poimirlem26  34449  dvnprodlem1  41772
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