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Theorem eliunxp 5292
Description: Membership in a union of Cartesian products. Analogue of elxp 5165 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 5160 . . . . . 6 Rel ({𝑥} × 𝐵)
21rgenw 2953 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
3 reliun 5272 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
42, 3mpbir 221 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
5 elrel 5256 . . . 4 ((Rel 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 706 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
76pm4.71ri 666 . 2 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
8 nfiu1 4582 . . . 4 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
98nfel2 2810 . . 3 𝑥 𝐶 𝑥𝐴 ({𝑥} × 𝐵)
10919.41 2141 . 2 (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
11 19.41v 1917 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
12 eleq1 2718 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
13 opeliunxp 5204 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
1412, 13syl6bb 276 . . . . . 6 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
1514pm5.32i 670 . . . . 5 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1615exbii 1814 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1711, 16bitr3i 266 . . 3 ((∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1817exbii 1814 . 2 (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
197, 10, 183bitr2i 288 1 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  {csn 4210  cop 4216   ciun 4552   × cxp 5141  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-iun 4554  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  raliunxp  5294  dfmpt3  6052  mpt2mptx  6793  fsumcom2  14549  fsumcom2OLD  14550  fprodcom2  14758  fprodcom2OLD  14759  isfunc  16571  gsum2d2  18419  dprd2d2  18489  fsumvma  24983  mpt2mptxf  29605  poimirlem26  33565  dvnprodlem1  40479
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