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Theorem eliunxp 4678
Description: Membership in a union of cross products. Analogue of elxp 4556 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 4648 . . . . . 6 Rel ({𝑥} × 𝐵)
21rgenw 2487 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
3 reliun 4660 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
42, 3mpbir 145 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
5 elrel 4641 . . . 4 ((Rel 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 420 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
76pm4.71ri 389 . 2 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
8 nfiu1 3843 . . . 4 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
98nfel2 2294 . . 3 𝑥 𝐶 𝑥𝐴 ({𝑥} × 𝐵)
10919.41 1664 . 2 (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
11 19.41v 1874 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
12 eleq1 2202 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
13 opeliunxp 4594 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
1412, 13syl6bb 195 . . . . . 6 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
1514pm5.32i 449 . . . . 5 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1615exbii 1584 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1711, 16bitr3i 185 . . 3 ((∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1817exbii 1584 . 2 (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
197, 10, 183bitr2i 207 1 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  wral 2416  {csn 3527  cop 3530   ciun 3813   × cxp 4537  Rel wrel 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-iun 3815  df-opab 3990  df-xp 4545  df-rel 4546
This theorem is referenced by:  raliunxp  4680  rexiunxp  4681  dfmpt3  5245  mpomptx  5862  fisumcom2  11219
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