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Theorem eliunxp2 42911
Description: Membership in a union of Cartesian products over its second component, analogous to eliunxp 5463. (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 5330 . . . . . . . 8 Rel (𝐴 × {𝑦})
21rgenw 3105 . . . . . . 7 𝑦𝐵 Rel (𝐴 × {𝑦})
3 reliun 5443 . . . . . . 7 (Rel 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∀𝑦𝐵 Rel (𝐴 × {𝑦}))
42, 3mpbir 223 . . . . . 6 Rel 𝑦𝐵 (𝐴 × {𝑦})
5 elrel 5426 . . . . . 6 ((Rel 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 682 . . . . 5 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7 excom 2205 . . . . 5 (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
86, 7sylibr 226 . . . 4 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩)
98pm4.71ri 557 . . 3 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
10 nfiu1 4740 . . . . 5 𝑦 𝑦𝐵 (𝐴 × {𝑦})
1110nfel2 2958 . . . 4 𝑦 𝐶 𝑦𝐵 (𝐴 × {𝑦})
121119.41 2270 . . 3 (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
13 19.41v 2045 . . . . 5 (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
14 eleq1 2866 . . . . . . . 8 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦})))
15 opeliun2xp 42910 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝑥𝐴))
16 ancom 453 . . . . . . . . 9 ((𝑦𝐵𝑥𝐴) ↔ (𝑥𝐴𝑦𝐵))
1715, 16bitri 267 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵))
1814, 17syl6bb 279 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵)))
1918pm5.32i 571 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2019exbii 1944 . . . . 5 (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2113, 20bitr3i 269 . . . 4 ((∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2221exbii 1944 . . 3 (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ ∃𝑦𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
239, 12, 223bitr2i 291 . 2 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑦𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
24 excom 2205 . 2 (∃𝑦𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2523, 24bitri 267 1 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  wral 3089  {csn 4368  cop 4374   ciun 4710   × cxp 5310  Rel wrel 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-iun 4712  df-opab 4906  df-xp 5318  df-rel 5319
This theorem is referenced by:  mpt2mptx2  42912
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