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Theorem eliunxp2 47088
Description: Membership in a union of Cartesian products over its second component, analogous to eliunxp 5837. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
eliunxp2 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦,𝐶
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem eliunxp2
StepHypRef Expression
1 relxp 5694 . . . . . . . 8 Rel (𝐴 × {𝑦})
21rgenw 3065 . . . . . . 7 𝑦𝐵 Rel (𝐴 × {𝑦})
3 reliun 5816 . . . . . . 7 (Rel 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∀𝑦𝐵 Rel (𝐴 × {𝑦}))
42, 3mpbir 230 . . . . . 6 Rel 𝑦𝐵 (𝐴 × {𝑦})
5 elrel 5798 . . . . . 6 ((Rel 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 688 . . . . 5 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7 excom 2162 . . . . 5 (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
86, 7sylibr 233 . . . 4 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩)
98pm4.71ri 561 . . 3 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
10 nfiu1 5031 . . . . 5 𝑦 𝑦𝐵 (𝐴 × {𝑦})
1110nfel2 2921 . . . 4 𝑦 𝐶 𝑦𝐵 (𝐴 × {𝑦})
121119.41 2228 . . 3 (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
13 19.41v 1953 . . . . 5 (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
14 eleq1 2821 . . . . . . . 8 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦})))
15 opeliun2xp 47087 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝑥𝐴))
1615biancomi 463 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵))
1714, 16bitrdi 286 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵)))
1817pm5.32i 575 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1918exbii 1850 . . . . 5 (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2013, 19bitr3i 276 . . . 4 ((∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2120exbii 1850 . . 3 (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ ∃𝑦𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
229, 12, 213bitr2i 298 . 2 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑦𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
23 excom 2162 . 2 (∃𝑦𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2422, 23bitri 274 1 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3061  {csn 4628  cop 4634   ciun 4997   × cxp 5674  Rel wrel 5681
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-iun 4999  df-opab 5211  df-xp 5682  df-rel 5683
This theorem is referenced by:  mpomptx2  47089
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