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Theorem opeliun2xp 41399
Description: Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5131. (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 4487 . . 3 𝑦𝐵 (𝐴 × {𝑦}) = {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})}
21eleq2i 2690 . 2 (⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ ⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})})
3 opex 4893 . . 3 𝐶, 𝑦⟩ ∈ V
4 df-rex 2913 . . . . 5 (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑦(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})))
5 nfv 1840 . . . . . 6 𝑧(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦}))
6 nfs1v 2436 . . . . . . 7 𝑦[𝑧 / 𝑦]𝑦𝐵
7 nfcsb1v 3530 . . . . . . . . 9 𝑦𝑧 / 𝑦𝐴
8 nfcv 2761 . . . . . . . . 9 𝑦{𝑧}
97, 8nfxp 5102 . . . . . . . 8 𝑦(𝑧 / 𝑦𝐴 × {𝑧})
109nfcri 2755 . . . . . . 7 𝑦 𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})
116, 10nfan 1825 . . . . . 6 𝑦([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}))
12 sbequ12 2108 . . . . . . 7 (𝑦 = 𝑧 → (𝑦𝐵 ↔ [𝑧 / 𝑦]𝑦𝐵))
13 csbeq1a 3523 . . . . . . . . 9 (𝑦 = 𝑧𝐴 = 𝑧 / 𝑦𝐴)
14 sneq 4158 . . . . . . . . 9 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1513, 14xpeq12d 5100 . . . . . . . 8 (𝑦 = 𝑧 → (𝐴 × {𝑦}) = (𝑧 / 𝑦𝐴 × {𝑧}))
1615eleq2d 2684 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 ∈ (𝐴 × {𝑦}) ↔ 𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
1712, 16anbi12d 746 . . . . . 6 (𝑦 = 𝑧 → ((𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})) ↔ ([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
185, 11, 17cbvex 2271 . . . . 5 (∃𝑦(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
194, 18bitri 264 . . . 4 (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
20 eleq1 2686 . . . . . 6 (𝑥 = ⟨𝐶, 𝑦⟩ → (𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}) ↔ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
2120anbi2d 739 . . . . 5 (𝑥 = ⟨𝐶, 𝑦⟩ → (([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
2221exbidv 1847 . . . 4 (𝑥 = ⟨𝐶, 𝑦⟩ → (∃𝑧([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
2319, 22syl5bb 272 . . 3 (𝑥 = ⟨𝐶, 𝑦⟩ → (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
243, 23elab 3333 . 2 (⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})} ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
25 opelxp 5106 . . . . . 6 (⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}) ↔ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧}))
2625anbi2i 729 . . . . 5 (([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})))
27 an13 839 . . . . . 6 (([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ (𝐶𝑧 / 𝑦𝐴 ∧ [𝑧 / 𝑦]𝑦𝐵)))
28 ancom 466 . . . . . . 7 ((𝐶𝑧 / 𝑦𝐴 ∧ [𝑧 / 𝑦]𝑦𝐵) ↔ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴))
2928anbi2i 729 . . . . . 6 ((𝑦 ∈ {𝑧} ∧ (𝐶𝑧 / 𝑦𝐴 ∧ [𝑧 / 𝑦]𝑦𝐵)) ↔ (𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
3027, 29bitri 264 . . . . 5 (([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
31 velsn 4164 . . . . . . 7 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
32 equcom 1942 . . . . . . 7 (𝑦 = 𝑧𝑧 = 𝑦)
3331, 32bitri 264 . . . . . 6 (𝑦 ∈ {𝑧} ↔ 𝑧 = 𝑦)
3433anbi1i 730 . . . . 5 ((𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)) ↔ (𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
3526, 30, 343bitri 286 . . . 4 (([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ (𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
3635exbii 1771 . . 3 (∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
37 vex 3189 . . . 4 𝑦 ∈ V
38 sbequ12r 2109 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝑦𝐵𝑦𝐵))
3913equcoms 1944 . . . . . . 7 (𝑧 = 𝑦𝐴 = 𝑧 / 𝑦𝐴)
4039eqcomd 2627 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑦𝐴 = 𝐴)
4140eleq2d 2684 . . . . 5 (𝑧 = 𝑦 → (𝐶𝑧 / 𝑦𝐴𝐶𝐴))
4238, 41anbi12d 746 . . . 4 (𝑧 = 𝑦 → (([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴) ↔ (𝑦𝐵𝐶𝐴)))
4337, 42ceqsexv 3228 . . 3 (∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)) ↔ (𝑦𝐵𝐶𝐴))
4436, 43bitri 264 . 2 (∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ (𝑦𝐵𝐶𝐴))
452, 24, 443bitri 286 1 (⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  [wsb 1877  wcel 1987  {cab 2607  wrex 2908  csb 3514  {csn 4148  cop 4154   ciun 4485   × cxp 5072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-iun 4487  df-opab 4674  df-xp 5080
This theorem is referenced by:  eliunxp2  41400
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