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Theorem opeliun2xp 44664
Description: Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5606. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
opeliun2xp (⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝐶𝐴))

Proof of Theorem opeliun2xp
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iun 4907 . . 3 𝑦𝐵 (𝐴 × {𝑦}) = {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})}
21eleq2i 2907 . 2 (⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ ⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})})
3 opex 5343 . . 3 𝐶, 𝑦⟩ ∈ V
4 df-rex 3139 . . . . 5 (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑦(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})))
5 nfv 1916 . . . . . 6 𝑧(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦}))
6 nfs1v 2161 . . . . . . 7 𝑦[𝑧 / 𝑦]𝑦𝐵
7 nfcsb1v 3890 . . . . . . . . 9 𝑦𝑧 / 𝑦𝐴
8 nfcv 2982 . . . . . . . . 9 𝑦{𝑧}
97, 8nfxp 5575 . . . . . . . 8 𝑦(𝑧 / 𝑦𝐴 × {𝑧})
109nfcri 2969 . . . . . . 7 𝑦 𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})
116, 10nfan 1901 . . . . . 6 𝑦([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}))
12 sbequ12 2255 . . . . . . 7 (𝑦 = 𝑧 → (𝑦𝐵 ↔ [𝑧 / 𝑦]𝑦𝐵))
13 csbeq1a 3880 . . . . . . . . 9 (𝑦 = 𝑧𝐴 = 𝑧 / 𝑦𝐴)
14 sneq 4560 . . . . . . . . 9 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1513, 14xpeq12d 5573 . . . . . . . 8 (𝑦 = 𝑧 → (𝐴 × {𝑦}) = (𝑧 / 𝑦𝐴 × {𝑧}))
1615eleq2d 2901 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 ∈ (𝐴 × {𝑦}) ↔ 𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
1712, 16anbi12d 633 . . . . . 6 (𝑦 = 𝑧 → ((𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})) ↔ ([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
185, 11, 17cbvexv1 2364 . . . . 5 (∃𝑦(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
194, 18bitri 278 . . . 4 (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
20 eleq1 2903 . . . . . 6 (𝑥 = ⟨𝐶, 𝑦⟩ → (𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}) ↔ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
2120anbi2d 631 . . . . 5 (𝑥 = ⟨𝐶, 𝑦⟩ → (([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
2221exbidv 1923 . . . 4 (𝑥 = ⟨𝐶, 𝑦⟩ → (∃𝑧([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
2319, 22syl5bb 286 . . 3 (𝑥 = ⟨𝐶, 𝑦⟩ → (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
243, 23elab 3653 . 2 (⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})} ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
25 opelxp 5578 . . . . . 6 (⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}) ↔ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧}))
2625anbi2i 625 . . . . 5 (([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})))
27 an13 646 . . . . . 6 (([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ (𝐶𝑧 / 𝑦𝐴 ∧ [𝑧 / 𝑦]𝑦𝐵)))
28 ancom 464 . . . . . . 7 ((𝐶𝑧 / 𝑦𝐴 ∧ [𝑧 / 𝑦]𝑦𝐵) ↔ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴))
2928anbi2i 625 . . . . . 6 ((𝑦 ∈ {𝑧} ∧ (𝐶𝑧 / 𝑦𝐴 ∧ [𝑧 / 𝑦]𝑦𝐵)) ↔ (𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
3027, 29bitri 278 . . . . 5 (([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
31 velsn 4566 . . . . . . 7 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
32 equcom 2026 . . . . . . 7 (𝑦 = 𝑧𝑧 = 𝑦)
3331, 32bitri 278 . . . . . 6 (𝑦 ∈ {𝑧} ↔ 𝑧 = 𝑦)
3433anbi1i 626 . . . . 5 ((𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)) ↔ (𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
3526, 30, 343bitri 300 . . . 4 (([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ (𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
3635exbii 1849 . . 3 (∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
37 sbequ12r 2256 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝑦𝐵𝑦𝐵))
3813equcoms 2028 . . . . . . 7 (𝑧 = 𝑦𝐴 = 𝑧 / 𝑦𝐴)
3938eqcomd 2830 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑦𝐴 = 𝐴)
4039eleq2d 2901 . . . . 5 (𝑧 = 𝑦 → (𝐶𝑧 / 𝑦𝐴𝐶𝐴))
4137, 40anbi12d 633 . . . 4 (𝑧 = 𝑦 → (([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴) ↔ (𝑦𝐵𝐶𝐴)))
4241equsexvw 2012 . . 3 (∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)) ↔ (𝑦𝐵𝐶𝐴))
4336, 42bitri 278 . 2 (∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ (𝑦𝐵𝐶𝐴))
442, 24, 433bitri 300 1 (⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  [wsb 2070  wcel 2115  {cab 2802  wrex 3134  csb 3866  {csn 4550  cop 4556   ciun 4905   × cxp 5540
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-iun 4907  df-opab 5115  df-xp 5548
This theorem is referenced by:  eliunxp2  44665
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