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Theorem opeliunxp 4463
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 2624 . 2 (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) → ⟨𝑥, 𝐶⟩ ∈ V)
2 opexg 4031 . 2 ((𝑥𝐴𝐶𝐵) → ⟨𝑥, 𝐶⟩ ∈ V)
3 df-rex 2361 . . . . . 6 (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)))
4 nfv 1464 . . . . . . 7 𝑧(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵))
5 nfs1v 1860 . . . . . . . 8 𝑥[𝑧 / 𝑥]𝑥𝐴
6 nfcv 2225 . . . . . . . . . 10 𝑥{𝑧}
7 nfcsb1v 2952 . . . . . . . . . 10 𝑥𝑧 / 𝑥𝐵
86, 7nfxp 4439 . . . . . . . . 9 𝑥({𝑧} × 𝑧 / 𝑥𝐵)
98nfcri 2219 . . . . . . . 8 𝑥 𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)
105, 9nfan 1500 . . . . . . 7 𝑥([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵))
11 sbequ12 1698 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐴 ↔ [𝑧 / 𝑥]𝑥𝐴))
12 sneq 3442 . . . . . . . . . 10 (𝑥 = 𝑧 → {𝑥} = {𝑧})
13 csbeq1a 2930 . . . . . . . . . 10 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
1412, 13xpeq12d 4438 . . . . . . . . 9 (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × 𝑧 / 𝑥𝐵))
1514eleq2d 2154 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 ∈ ({𝑥} × 𝐵) ↔ 𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
1611, 15anbi12d 457 . . . . . . 7 (𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
174, 10, 16cbvex 1683 . . . . . 6 (∃𝑥(𝑥𝐴𝑦 ∈ ({𝑥} × 𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
183, 17bitri 182 . . . . 5 (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
19 eleq1 2147 . . . . . . 7 (𝑦 = ⟨𝑥, 𝐶⟩ → (𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵) ↔ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)))
2019anbi2d 452 . . . . . 6 (𝑦 = ⟨𝑥, 𝐶⟩ → (([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
2120exbidv 1750 . . . . 5 (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑧([𝑧 / 𝑥]𝑥𝐴𝑦 ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
2218, 21syl5bb 190 . . . 4 (𝑦 = ⟨𝑥, 𝐶⟩ → (∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
23 df-iun 3717 . . . 4 𝑥𝐴 ({𝑥} × 𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ ({𝑥} × 𝐵)}
2422, 23elab2g 2753 . . 3 (⟨𝑥, 𝐶⟩ ∈ V → (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵))))
25 opelxp 4442 . . . . . . 7 (⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵) ↔ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵))
2625anbi2i 445 . . . . . 6 (([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵)))
27 an12 526 . . . . . 6 (([𝑧 / 𝑥]𝑥𝐴 ∧ (𝑥 ∈ {𝑧} ∧ 𝐶𝑧 / 𝑥𝐵)) ↔ (𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
28 velsn 3448 . . . . . . . 8 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
29 equcom 1637 . . . . . . . 8 (𝑥 = 𝑧𝑧 = 𝑥)
3028, 29bitri 182 . . . . . . 7 (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
3130anbi1i 446 . . . . . 6 ((𝑥 ∈ {𝑧} ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
3226, 27, 313bitri 204 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ (𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
3332exbii 1539 . . . 4 (∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)))
34 vex 2618 . . . . 5 𝑥 ∈ V
35 sbequ12r 1699 . . . . . 6 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝑥𝐴𝑥𝐴))
3613equcoms 1638 . . . . . . . 8 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3736eqcomd 2090 . . . . . . 7 (𝑧 = 𝑥𝑧 / 𝑥𝐵 = 𝐵)
3837eleq2d 2154 . . . . . 6 (𝑧 = 𝑥 → (𝐶𝑧 / 𝑥𝐵𝐶𝐵))
3935, 38anbi12d 457 . . . . 5 (𝑧 = 𝑥 → (([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵) ↔ (𝑥𝐴𝐶𝐵)))
4034, 39ceqsexv 2652 . . . 4 (∃𝑧(𝑧 = 𝑥 ∧ ([𝑧 / 𝑥]𝑥𝐴𝐶𝑧 / 𝑥𝐵)) ↔ (𝑥𝐴𝐶𝐵))
4133, 40bitri 182 . . 3 (∃𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ ⟨𝑥, 𝐶⟩ ∈ ({𝑧} × 𝑧 / 𝑥𝐵)) ↔ (𝑥𝐴𝐶𝐵))
4224, 41syl6bb 194 . 2 (⟨𝑥, 𝐶⟩ ∈ V → (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵)))
431, 2, 42pm5.21nii 653 1 (⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  [wsb 1689  wrex 2356  Vcvv 2615  csb 2922  {csn 3431  cop 3434   ciun 3715   × cxp 4411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-iun 3717  df-opab 3877  df-xp 4419
This theorem is referenced by:  eliunxp  4545  opeliunxp2  4546
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