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Theorem opeliun2xp 41904
Description: Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5093. (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 4457 . . 3 𝑦𝐵 (𝐴 × {𝑦}) = {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})}
21eleq2i 2680 . 2 (⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ ⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})})
3 opex 4859 . . 3 𝐶, 𝑦⟩ ∈ V
4 df-rex 2902 . . . . 5 (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑦(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})))
5 nfv 1830 . . . . . 6 𝑧(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦}))
6 nfs1v 2425 . . . . . . 7 𝑦[𝑧 / 𝑦]𝑦𝐵
7 nfcsb1v 3515 . . . . . . . . 9 𝑦𝑧 / 𝑦𝐴
8 nfcv 2751 . . . . . . . . 9 𝑦{𝑧}
97, 8nfxp 5066 . . . . . . . 8 𝑦(𝑧 / 𝑦𝐴 × {𝑧})
109nfcri 2745 . . . . . . 7 𝑦 𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})
116, 10nfan 1816 . . . . . 6 𝑦([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}))
12 sbequ12 2097 . . . . . . 7 (𝑦 = 𝑧 → (𝑦𝐵 ↔ [𝑧 / 𝑦]𝑦𝐵))
13 csbeq1a 3508 . . . . . . . . 9 (𝑦 = 𝑧𝐴 = 𝑧 / 𝑦𝐴)
14 sneq 4135 . . . . . . . . 9 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1513, 14xpeq12d 5064 . . . . . . . 8 (𝑦 = 𝑧 → (𝐴 × {𝑦}) = (𝑧 / 𝑦𝐴 × {𝑧}))
1615eleq2d 2673 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 ∈ (𝐴 × {𝑦}) ↔ 𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
1712, 16anbi12d 743 . . . . . 6 (𝑦 = 𝑧 → ((𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})) ↔ ([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
185, 11, 17cbvex 2260 . . . . 5 (∃𝑦(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
194, 18bitri 263 . . . 4 (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
20 eleq1 2676 . . . . . 6 (𝑥 = ⟨𝐶, 𝑦⟩ → (𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}) ↔ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
2120anbi2d 736 . . . . 5 (𝑥 = ⟨𝐶, 𝑦⟩ → (([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
2221exbidv 1837 . . . 4 (𝑥 = ⟨𝐶, 𝑦⟩ → (∃𝑧([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
2319, 22syl5bb 271 . . 3 (𝑥 = ⟨𝐶, 𝑦⟩ → (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
243, 23elab 3319 . 2 (⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})} ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
25 opelxp 5070 . . . . . 6 (⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}) ↔ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧}))
2625anbi2i 726 . . . . 5 (([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})))
27 an13 836 . . . . . 6 (([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ (𝐶𝑧 / 𝑦𝐴 ∧ [𝑧 / 𝑦]𝑦𝐵)))
28 ancom 465 . . . . . . 7 ((𝐶𝑧 / 𝑦𝐴 ∧ [𝑧 / 𝑦]𝑦𝐵) ↔ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴))
2928anbi2i 726 . . . . . 6 ((𝑦 ∈ {𝑧} ∧ (𝐶𝑧 / 𝑦𝐴 ∧ [𝑧 / 𝑦]𝑦𝐵)) ↔ (𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
3027, 29bitri 263 . . . . 5 (([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
31 velsn 4141 . . . . . . 7 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
32 equcom 1932 . . . . . . 7 (𝑦 = 𝑧𝑧 = 𝑦)
3331, 32bitri 263 . . . . . 6 (𝑦 ∈ {𝑧} ↔ 𝑧 = 𝑦)
3433anbi1i 727 . . . . 5 ((𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)) ↔ (𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
3526, 30, 343bitri 285 . . . 4 (([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ (𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
3635exbii 1764 . . 3 (∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
37 vex 3176 . . . 4 𝑦 ∈ V
38 sbequ12r 2098 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝑦𝐵𝑦𝐵))
3913equcoms 1934 . . . . . . 7 (𝑧 = 𝑦𝐴 = 𝑧 / 𝑦𝐴)
4039eqcomd 2616 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑦𝐴 = 𝐴)
4140eleq2d 2673 . . . . 5 (𝑧 = 𝑦 → (𝐶𝑧 / 𝑦𝐴𝐶𝐴))
4238, 41anbi12d 743 . . . 4 (𝑧 = 𝑦 → (([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴) ↔ (𝑦𝐵𝐶𝐴)))
4337, 42ceqsexv 3215 . . 3 (∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)) ↔ (𝑦𝐵𝐶𝐴))
4436, 43bitri 263 . 2 (∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ (𝑦𝐵𝐶𝐴))
452, 24, 443bitri 285 1 (⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wex 1695  [wsb 1867  wcel 1977  {cab 2596  wrex 2897  csb 3499  {csn 4125  cop 4131   ciun 4455   × cxp 5036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-iun 4457  df-opab 4644  df-xp 5044
This theorem is referenced by:  eliunxp2  41905
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