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Theorem opeliunxp 5703
Description: Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
Assertion
Ref Expression
opeliunxp (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵))

Proof of Theorem opeliunxp
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iun 4960 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵)}
21eleq2i 2826 . 2 (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵)})
3 opex 5425 . . 3 𝑥, 𝐶⟩ ∈ V
4 df-rex 3071 . . . . 5 (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)))
5 nfv 1918 . . . . . 6 𝑧(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵))
6 nfs1v 2154 . . . . . . 7 𝑥[𝑧 / 𝑥]𝑥𝐴
7 nfcv 2904 . . . . . . . . 9 𝑥{𝑧}
8 nfcsb1v 3884 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐵
97, 8nfxp 5670 . . . . . . . 8 𝑥({𝑧} × 𝑧 / 𝑥𝐵)
109nfcri 2891 . . . . . . 7 𝑥 𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)
116, 10nfan 1903 . . . . . 6 𝑥([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵))
12 sbequ12 2244 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝐴 ↔ [𝑧 / 𝑥]𝑥𝐴))
13 sneq 4600 . . . . . . . . 9 (𝑥 = 𝑧 → {𝑥} = {𝑧})
14 csbeq1a 3873 . . . . . . . . 9 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
1513, 14xpeq12d 5668 . . . . . . . 8 (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × 𝑧 / 𝑥𝐵))
1615eleq2d 2820 . . . . . . 7 (𝑥 = 𝑧 → (𝑦 ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
1712, 16anbi12d 632 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
185, 11, 17cbvexv1 2339 . . . . 5 (∃𝑥(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
194, 18bitri 275 . . . 4 (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
20 eleq1 2822 . . . . . 6 (𝑦 = ⟨𝑥, 𝐶⟩ → (𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
2120anbi2d 630 . . . . 5 (𝑦 = ⟨𝑥, 𝐶⟩ → (([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
2221exbidv 1925 . . . 4 (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
2319, 22bitrid 283 . . 3 (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
243, 23elab 3634 . 2 (⟨𝑥, 𝐶⟩ ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵)} ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
25 opelxp 5673 . . . . . 6 (⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵) ↔ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵))
2625anbi2i 624 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵)))
27 an12 644 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵)) ↔ (𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
28 velsn 4606 . . . . . . 7 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
29 equcom 2022 . . . . . . 7 (𝑥 = 𝑧𝑧 = 𝑥)
3028, 29bitri 275 . . . . . 6 (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
3130anbi1i 625 . . . . 5 ((𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
3226, 27, 313bitri 297 . . . 4 (([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
3332exbii 1851 . . 3 (∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
34 sbequ12r 2245 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝑥𝐴𝑥𝐴))
3514equcoms 2024 . . . . . . 7 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3635eqcomd 2739 . . . . . 6 (𝑧 = 𝑥𝑧 / 𝑥𝐵 = 𝐵)
3736eleq2d 2820 . . . . 5 (𝑧 = 𝑥 → (𝐶𝑧 / 𝑥𝐵𝐶𝐵))
3834, 37anbi12d 632 . . . 4 (𝑧 = 𝑥 → (([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵) ↔ (𝑥𝐴𝐶𝐵)))
3938equsexvw 2009 . . 3 (∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)) ↔ (𝑥𝐴𝐶𝐵))
4033, 39bitri 275 . 2 (∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ (𝑥𝐴𝐶𝐵))
412, 24, 403bitri 297 1 (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  [wsb 2068  wcel 2107  {cab 2710  wrex 3070  csb 3859  {csn 4590  cop 4596   ciun 4958   × cxp 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-iun 4960  df-opab 5172  df-xp 5643
This theorem is referenced by:  eliunxp  5797  opeliunxp2  5798  opeliunxp2f  8145  gsum2d2lem  19758  gsum2d2  19759  gsumcom2  19760  dprdval  19790  ptbasfi  22955  cnextfun  23438  cnextfvval  23439  cnextf  23440  dvbsss  25289  iunsnima  31590
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