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Theorem opeliun2xp 42639
 Description: Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5327. (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 4674 . . 3 𝑦𝐵 (𝐴 × {𝑦}) = {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})}
21eleq2i 2831 . 2 (⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ ⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})})
3 opex 5081 . . 3 𝐶, 𝑦⟩ ∈ V
4 df-rex 3056 . . . . 5 (∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑦(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})))
5 nfv 1992 . . . . . 6 𝑧(𝑦𝐵𝑥 ∈ (𝐴 × {𝑦}))
6 nfs1v 2574 . . . . . . 7 𝑦[𝑧 / 𝑦]𝑦𝐵
7 nfcsb1v 3690 . . . . . . . . 9 𝑦𝑧 / 𝑦𝐴
8 nfcv 2902 . . . . . . . . 9 𝑦{𝑧}
97, 8nfxp 5299 . . . . . . . 8 𝑦(𝑧 / 𝑦𝐴 × {𝑧})
109nfcri 2896 . . . . . . 7 𝑦 𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})
116, 10nfan 1977 . . . . . 6 𝑦([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}))
12 sbequ12 2258 . . . . . . 7 (𝑦 = 𝑧 → (𝑦𝐵 ↔ [𝑧 / 𝑦]𝑦𝐵))
13 csbeq1a 3683 . . . . . . . . 9 (𝑦 = 𝑧𝐴 = 𝑧 / 𝑦𝐴)
14 sneq 4331 . . . . . . . . 9 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1513, 14xpeq12d 5297 . . . . . . . 8 (𝑦 = 𝑧 → (𝐴 × {𝑦}) = (𝑧 / 𝑦𝐴 × {𝑧}))
1615eleq2d 2825 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 ∈ (𝐴 × {𝑦}) ↔ 𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
1712, 16anbi12d 749 . . . . . 6 (𝑦 = 𝑧 → ((𝑦𝐵𝑥 ∈ (𝐴 × {𝑦})) ↔ ([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
185, 11, 17cbvex 2417 . . . . 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 2827 . . . . . 6 (𝑥 = ⟨𝐶, 𝑦⟩ → (𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧}) ↔ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
2120anbi2d 742 . . . . 5 (𝑥 = ⟨𝐶, 𝑦⟩ → (([𝑧 / 𝑦]𝑦𝐵𝑥 ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}))))
2221exbidv 1999 . . . 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 3490 . 2 (⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦𝐵 𝑥 ∈ (𝐴 × {𝑦})} ↔ ∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})))
25 opelxp 5303 . . . . . 6 (⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧}) ↔ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧}))
2625anbi2i 732 . . . . 5 (([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦𝐵 ∧ (𝐶𝑧 / 𝑦𝐴𝑦 ∈ {𝑧})))
27 an13 875 . . . . . 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 732 . . . . . 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 4337 . . . . . . 7 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
32 equcom 2100 . . . . . . 7 (𝑦 = 𝑧𝑧 = 𝑦)
3331, 32bitri 264 . . . . . 6 (𝑦 ∈ {𝑧} ↔ 𝑧 = 𝑦)
3433anbi1i 733 . . . . 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 1923 . . 3 (∃𝑧([𝑧 / 𝑦]𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (𝑧 / 𝑦𝐴 × {𝑧})) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴)))
37 vex 3343 . . . 4 𝑦 ∈ V
38 sbequ12r 2259 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝑦𝐵𝑦𝐵))
3913equcoms 2102 . . . . . . 7 (𝑧 = 𝑦𝐴 = 𝑧 / 𝑦𝐴)
4039eqcomd 2766 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑦𝐴 = 𝐴)
4140eleq2d 2825 . . . . 5 (𝑧 = 𝑦 → (𝐶𝑧 / 𝑦𝐴𝐶𝐴))
4238, 41anbi12d 749 . . . 4 (𝑧 = 𝑦 → (([𝑧 / 𝑦]𝑦𝐵𝐶𝑧 / 𝑦𝐴) ↔ (𝑦𝐵𝐶𝐴)))
4337, 42ceqsexv 3382 . . 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 383   = wceq 1632  ∃wex 1853  [wsb 2046   ∈ wcel 2139  {cab 2746  ∃wrex 3051  ⦋csb 3674  {csn 4321  ⟨cop 4327  ∪ ciun 4672   × cxp 5264 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-iun 4674  df-opab 4865  df-xp 5272 This theorem is referenced by:  eliunxp2  42640
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