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Theorem opeliunxp 4420
Description: Membership in a union of cross 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 elex 2581 . 2 (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) → ⟨𝑥, 𝐶⟩ ∈ V)
2 vex 2575 . . 3 𝑥 ∈ V
3 elex 2581 . . . 4 (𝐶𝐵𝐶 ∈ V)
43adantl 266 . . 3 ((𝑥𝐴𝐶𝐵) → 𝐶 ∈ V)
5 opexgOLD 3990 . . 3 ((𝑥 ∈ V ∧ 𝐶 ∈ V) → ⟨𝑥, 𝐶⟩ ∈ V)
62, 4, 5sylancr 399 . 2 ((𝑥𝐴𝐶𝐵) → ⟨𝑥, 𝐶⟩ ∈ V)
7 df-rex 2327 . . . . . 6 (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)))
8 nfv 1435 . . . . . . 7 𝑧(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵))
9 nfs1v 1829 . . . . . . . 8 𝑥[𝑧 / 𝑥]𝑥𝐴
10 nfcv 2192 . . . . . . . . . 10 𝑥{𝑧}
11 nfcsb1v 2907 . . . . . . . . . 10 𝑥𝑧 / 𝑥𝐵
1210, 11nfxp 4396 . . . . . . . . 9 𝑥({𝑧} × 𝑧 / 𝑥𝐵)
1312nfcri 2186 . . . . . . . 8 𝑥 𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)
149, 13nfan 1471 . . . . . . 7 𝑥([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵))
15 sbequ12 1668 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐴 ↔ [𝑧 / 𝑥]𝑥𝐴))
16 sneq 3411 . . . . . . . . . 10 (𝑥 = 𝑧 → {𝑥} = {𝑧})
17 csbeq1a 2885 . . . . . . . . . 10 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
1816, 17xpeq12d 4395 . . . . . . . . 9 (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × 𝑧 / 𝑥𝐵))
1918eleq2d 2121 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
2015, 19anbi12d 450 . . . . . . 7 (𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
218, 14, 20cbvex 1653 . . . . . 6 (∃𝑥(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
227, 21bitri 177 . . . . 5 (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
23 eleq1 2114 . . . . . . 7 (𝑦 = ⟨𝑥, 𝐶⟩ → (𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
2423anbi2d 445 . . . . . 6 (𝑦 = ⟨𝑥, 𝐶⟩ → (([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
2524exbidv 1720 . . . . 5 (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
2622, 25syl5bb 185 . . . 4 (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
27 df-iun 3684 . . . 4 𝑥𝐴 ({𝑥} × 𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵)}
2826, 27elab2g 2709 . . 3 (⟨𝑥, 𝐶⟩ ∈ V → (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
29 opelxp 4399 . . . . . . 7 (⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵) ↔ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵))
3029anbi2i 438 . . . . . 6 (([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵)))
31 an12 503 . . . . . 6 (([𝑧 / 𝑥]𝑥𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵)) ↔ (𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
32 velsn 3417 . . . . . . . 8 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
33 equcom 1607 . . . . . . . 8 (𝑥 = 𝑧𝑧 = 𝑥)
3432, 33bitri 177 . . . . . . 7 (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
3534anbi1i 439 . . . . . 6 ((𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
3630, 31, 353bitri 199 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
3736exbii 1510 . . . 4 (∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
38 sbequ12r 1669 . . . . . 6 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝑥𝐴𝑥𝐴))
3917equcoms 1608 . . . . . . . 8 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
4039eqcomd 2059 . . . . . . 7 (𝑧 = 𝑥𝑧 / 𝑥𝐵 = 𝐵)
4140eleq2d 2121 . . . . . 6 (𝑧 = 𝑥 → (𝐶𝑧 / 𝑥𝐵𝐶𝐵))
4238, 41anbi12d 450 . . . . 5 (𝑧 = 𝑥 → (([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵) ↔ (𝑥𝐴𝐶𝐵)))
432, 42ceqsexv 2608 . . . 4 (∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)) ↔ (𝑥𝐴𝐶𝐵))
4437, 43bitri 177 . . 3 (∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ (𝑥𝐴𝐶𝐵))
4528, 44syl6bb 189 . 2 (⟨𝑥, 𝐶⟩ ∈ V → (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵)))
461, 6, 45pm5.21nii 628 1 (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1257  wex 1395  wcel 1407  [wsb 1659  wrex 2322  Vcvv 2572  csb 2877  {csn 3400  cop 3403   ciun 3682   × cxp 4368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-sbc 2785  df-csb 2878  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-iun 3684  df-opab 3844  df-xp 4376
This theorem is referenced by:  eliunxp  4500  opeliunxp2  4501
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