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Theorem eliunxp 5676
 Description: Membership in a union of Cartesian products. Analogue of elxp 5546 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 5541 . . . . . 6 Rel ({𝑥} × 𝐵)
21rgenw 3118 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
3 reliun 5657 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
42, 3mpbir 234 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
5 elrel 5639 . . . 4 ((Rel 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 689 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
76pm4.71ri 564 . 2 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
8 nfiu1 4919 . . . 4 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
98nfel2 2973 . . 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 1950 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)))
12 eleq1 2877 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
13 opeliunxp 5587 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
1412, 13syl6bb 290 . . . . . 6 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
1514pm5.32i 578 . . . . 5 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1615exbii 1849 . . . 4 (∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1711, 16bitr3i 280 . . 3 ((∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1817exbii 1849 . 2 (∃𝑥(∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑥𝐴 ({𝑥} × 𝐵)) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
197, 10, 183bitr2i 302 1 (𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  {csn 4528  ⟨cop 4534  ∪ ciun 4885   × cxp 5521  Rel wrel 5528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-iun 4887  df-opab 5097  df-xp 5529  df-rel 5530 This theorem is referenced by:  raliunxp  5678  dfmpt3  6462  mpomptx  7254  fsumcom2  15141  fprodcom2  15350  isfunc  17146  gsum2d2  19108  dprd2d2  19180  fsumvma  25841  2ndresdju  30455  mpomptxf  30486  poimirlem26  35234  dvnprodlem1  42756
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