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Theorem eliunxp2 47099
Description: Membership in a union of Cartesian products over its second component, analogous to eliunxp 5838. (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 5695 . . . . . . . 8 Rel (𝐴 × {𝑦})
21rgenw 3064 . . . . . . 7 𝑦𝐵 Rel (𝐴 × {𝑦})
3 reliun 5817 . . . . . . 7 (Rel 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∀𝑦𝐵 Rel (𝐴 × {𝑦}))
42, 3mpbir 230 . . . . . 6 Rel 𝑦𝐵 (𝐴 × {𝑦})
5 elrel 5799 . . . . . 6 ((Rel 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 687 . . . . 5 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7 excom 2161 . . . . 5 (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
86, 7sylibr 233 . . . 4 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩)
98pm4.71ri 560 . . 3 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
10 nfiu1 5032 . . . . 5 𝑦 𝑦𝐵 (𝐴 × {𝑦})
1110nfel2 2920 . . . 4 𝑦 𝐶 𝑦𝐵 (𝐴 × {𝑦})
121119.41 2227 . . 3 (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
13 19.41v 1952 . . . . 5 (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
14 eleq1 2820 . . . . . . . 8 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦})))
15 opeliun2xp 47098 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝑥𝐴))
1615biancomi 462 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵))
1714, 16bitrdi 286 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵)))
1817pm5.32i 574 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1918exbii 1849 . . . . 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 1849 . . 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 2161 . 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 395   = wceq 1540  wex 1780  wcel 2105  wral 3060  {csn 4629  cop 4635   ciun 4998   × cxp 5675  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 5000  df-opab 5212  df-xp 5683  df-rel 5684
This theorem is referenced by:  mpomptx2  47100
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