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Theorem opeliunxp 5136
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 4492 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵)}
21eleq2i 2690 . 2 (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵)})
3 opex 4898 . . 3 𝑥, 𝐶⟩ ∈ V
4 df-rex 2913 . . . . 5 (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)))
5 nfv 1840 . . . . . 6 𝑧(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵))
6 nfs1v 2436 . . . . . . 7 𝑥[𝑧 / 𝑥]𝑥𝐴
7 nfcv 2761 . . . . . . . . 9 𝑥{𝑧}
8 nfcsb1v 3534 . . . . . . . . 9 𝑥𝑧 / 𝑥𝐵
97, 8nfxp 5107 . . . . . . . 8 𝑥({𝑧} × 𝑧 / 𝑥𝐵)
109nfcri 2755 . . . . . . 7 𝑥 𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)
116, 10nfan 1825 . . . . . 6 𝑥([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵))
12 sbequ12 2108 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝐴 ↔ [𝑧 / 𝑥]𝑥𝐴))
13 sneq 4163 . . . . . . . . 9 (𝑥 = 𝑧 → {𝑥} = {𝑧})
14 csbeq1a 3527 . . . . . . . . 9 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
1513, 14xpeq12d 5105 . . . . . . . 8 (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × 𝑧 / 𝑥𝐵))
1615eleq2d 2684 . . . . . . 7 (𝑥 = 𝑧 → (𝑦 ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
1712, 16anbi12d 746 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
185, 11, 17cbvex 2271 . . . . 5 (∃𝑥(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
194, 18bitri 264 . . . 4 (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
20 eleq1 2686 . . . . . 6 (𝑦 = ⟨𝑥, 𝐶⟩ → (𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
2120anbi2d 739 . . . . 5 (𝑦 = ⟨𝑥, 𝐶⟩ → (([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
2221exbidv 1847 . . . 4 (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
2319, 22syl5bb 272 . . 3 (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
243, 23elab 3337 . 2 (⟨𝑥, 𝐶⟩ ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵)} ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
25 opelxp 5111 . . . . . 6 (⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵) ↔ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵))
2625anbi2i 729 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵)))
27 an12 837 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵)) ↔ (𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
28 velsn 4169 . . . . . . 7 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
29 equcom 1942 . . . . . . 7 (𝑥 = 𝑧𝑧 = 𝑥)
3028, 29bitri 264 . . . . . 6 (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
3130anbi1i 730 . . . . 5 ((𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
3226, 27, 313bitri 286 . . . 4 (([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
3332exbii 1771 . . 3 (∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
34 sbequ12r 2109 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝑥𝐴𝑥𝐴))
3514equcoms 1944 . . . . . . 7 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3635eqcomd 2627 . . . . . 6 (𝑧 = 𝑥𝑧 / 𝑥𝐵 = 𝐵)
3736eleq2d 2684 . . . . 5 (𝑧 = 𝑥 → (𝐶𝑧 / 𝑥𝐵𝐶𝐵))
3834, 37anbi12d 746 . . . 4 (𝑧 = 𝑥 → (([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵) ↔ (𝑥𝐴𝐶𝐵)))
3938equsexvw 1929 . . 3 (∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)) ↔ (𝑥𝐴𝐶𝐵))
4033, 39bitri 264 . 2 (∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ (𝑥𝐴𝐶𝐵))
412, 24, 403bitri 286 1 (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵))
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 3518  {csn 4153  cop 4159   ciun 4490   × cxp 5077
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 4746  ax-nul 4754  ax-pr 4872
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-iun 4492  df-opab 4679  df-xp 5085
This theorem is referenced by:  eliunxp  5224  opeliunxp2  5225  opeliunxp2f  7288  gsum2d2lem  18304  gsum2d2  18305  gsumcom2  18306  dprdval  18334  ptbasfi  21307  cnextfun  21791  cnextfvval  21792  cnextf  21793  dvbsss  23589  iunsnima  29294
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