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Theorem opabex3d 7808
Description: Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 9-Aug-2024.)
Hypotheses
Ref Expression
opabex3d.1 (𝜑𝐴𝑉)
opabex3d.2 ((𝜑𝑥𝐴) → {𝑦𝜓} ∈ V)
Assertion
Ref Expression
opabex3d (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ∈ V)
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem opabex3d
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.42v 1957 . . . . . 6 (∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2 an12 642 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)) ↔ (𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
32exbii 1850 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4 elxp 5612 . . . . . . . 8 (𝑧 ∈ ({𝑥} × {𝑦𝜓}) ↔ ∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})))
5 excom 2162 . . . . . . . . 9 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})))
6 an12 642 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
7 velsn 4577 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
87anbi1i 624 . . . . . . . . . . . . 13 ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
96, 8bitri 274 . . . . . . . . . . . 12 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
109exbii 1850 . . . . . . . . . . 11 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
11 opeq1 4804 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
1211eqeq2d 2749 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
1312anbi1d 630 . . . . . . . . . . . 12 (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
1413equsexvw 2008 . . . . . . . . . . 11 (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
1510, 14bitri 274 . . . . . . . . . 10 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
1615exbii 1850 . . . . . . . . 9 (∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
175, 16bitri 274 . . . . . . . 8 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
18 nfv 1917 . . . . . . . . . 10 𝑦 𝑧 = ⟨𝑥, 𝑤
19 nfsab1 2723 . . . . . . . . . 10 𝑦 𝑤 ∈ {𝑦𝜓}
2018, 19nfan 1902 . . . . . . . . 9 𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})
21 nfv 1917 . . . . . . . . 9 𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
22 opeq2 4805 . . . . . . . . . . 11 (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
2322eqeq2d 2749 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
24 df-clab 2716 . . . . . . . . . . 11 (𝑤 ∈ {𝑦𝜓} ↔ [𝑤 / 𝑦]𝜓)
25 sbequ12 2244 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
2625equcoms 2023 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
2724, 26bitr4id 290 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤 ∈ {𝑦𝜓} ↔ 𝜓))
2823, 27anbi12d 631 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2920, 21, 28cbvexv1 2339 . . . . . . . 8 (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
304, 17, 293bitri 297 . . . . . . 7 (𝑧 ∈ ({𝑥} × {𝑦𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
3130anbi2i 623 . . . . . 6 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
321, 3, 313bitr4ri 304 . . . . 5 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)))
3332exbii 1850 . . . 4 (∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)))
34 eliun 4928 . . . . 5 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜓}))
35 df-rex 3070 . . . . 5 (∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜓}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})))
3634, 35bitri 274 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})))
37 elopab 5440 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)))
3833, 36, 373bitr4i 303 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)})
3938eqriv 2735 . 2 𝑥𝐴 ({𝑥} × {𝑦𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)}
40 opabex3d.1 . . 3 (𝜑𝐴𝑉)
41 snex 5354 . . . . 5 {𝑥} ∈ V
42 opabex3d.2 . . . . 5 ((𝜑𝑥𝐴) → {𝑦𝜓} ∈ V)
43 xpexg 7600 . . . . 5 (({𝑥} ∈ V ∧ {𝑦𝜓} ∈ V) → ({𝑥} × {𝑦𝜓}) ∈ V)
4441, 42, 43sylancr 587 . . . 4 ((𝜑𝑥𝐴) → ({𝑥} × {𝑦𝜓}) ∈ V)
4544ralrimiva 3103 . . 3 (𝜑 → ∀𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V)
46 iunexg 7806 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V) → 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V)
4740, 45, 46syl2anc 584 . 2 (𝜑 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V)
4839, 47eqeltrrid 2844 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wex 1782  [wsb 2067  wcel 2106  {cab 2715  wral 3064  wrex 3065  Vcvv 3432  {csn 4561  cop 4567   ciun 4924  {copab 5136   × cxp 5587
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-opab 5137  df-xp 5595  df-rel 5596
This theorem is referenced by:  wksfval  27976  fpwrelmap  31068  satfv0  33320  cnvepresex  36469  opabresex0d  44777  upwlksfval  45297
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