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Theorem opabex3d 5748
 Description: Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
Hypotheses
Ref Expression
opabex3d.1 (𝜑𝐴 ∈ V)
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)

Proof of Theorem opabex3d
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.42v 1786 . . . . . 6 (∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2 an12 495 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)) ↔ (𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
32exbii 1496 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4 elxp 4362 . . . . . . . 8 (𝑧 ∈ ({𝑥} × {𝑦𝜓}) ↔ ∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})))
5 excom 1554 . . . . . . . . 9 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})))
6 an12 495 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
7 velsn 3392 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
87anbi1i 431 . . . . . . . . . . . . 13 ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
96, 8bitri 173 . . . . . . . . . . . 12 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
109exbii 1496 . . . . . . . . . . 11 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
11 vex 2560 . . . . . . . . . . . 12 𝑥 ∈ V
12 opeq1 3549 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
1312eqeq2d 2051 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
1413anbi1d 438 . . . . . . . . . . . 12 (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
1511, 14ceqsexv 2593 . . . . . . . . . . 11 (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
1610, 15bitri 173 . . . . . . . . . 10 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
1716exbii 1496 . . . . . . . . 9 (∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
185, 17bitri 173 . . . . . . . 8 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
19 nfv 1421 . . . . . . . . . 10 𝑦 𝑧 = ⟨𝑥, 𝑤
20 nfsab1 2030 . . . . . . . . . 10 𝑦 𝑤 ∈ {𝑦𝜓}
2119, 20nfan 1457 . . . . . . . . 9 𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})
22 nfv 1421 . . . . . . . . 9 𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
23 opeq2 3550 . . . . . . . . . . 11 (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
2423eqeq2d 2051 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
25 sbequ12 1654 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
2625equcoms 1594 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
27 df-clab 2027 . . . . . . . . . . 11 (𝑤 ∈ {𝑦𝜓} ↔ [𝑤 / 𝑦]𝜓)
2826, 27syl6rbbr 188 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤 ∈ {𝑦𝜓} ↔ 𝜓))
2924, 28anbi12d 442 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3021, 22, 29cbvex 1639 . . . . . . . 8 (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
314, 18, 303bitri 195 . . . . . . 7 (𝑧 ∈ ({𝑥} × {𝑦𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
3231anbi2i 430 . . . . . 6 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
331, 3, 323bitr4ri 202 . . . . 5 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)))
3433exbii 1496 . . . 4 (∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)))
35 eliun 3661 . . . . 5 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜓}))
36 df-rex 2312 . . . . 5 (∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜓}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})))
3735, 36bitri 173 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})))
38 elopab 3995 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)))
3934, 37, 383bitr4i 201 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)})
4039eqriv 2037 . 2 𝑥𝐴 ({𝑥} × {𝑦𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)}
41 opabex3d.1 . . 3 (𝜑𝐴 ∈ V)
42 snexg 3936 . . . . . 6 (𝑥 ∈ V → {𝑥} ∈ V)
4311, 42ax-mp 7 . . . . 5 {𝑥} ∈ V
44 opabex3d.2 . . . . 5 ((𝜑𝑥𝐴) → {𝑦𝜓} ∈ V)
45 xpexg 4452 . . . . 5 (({𝑥} ∈ V ∧ {𝑦𝜓} ∈ V) → ({𝑥} × {𝑦𝜓}) ∈ V)
4643, 44, 45sylancr 393 . . . 4 ((𝜑𝑥𝐴) → ({𝑥} × {𝑦𝜓}) ∈ V)
4746ralrimiva 2392 . . 3 (𝜑 → ∀𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V)
48 iunexg 5746 . . 3 ((𝐴 ∈ V ∧ ∀𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V) → 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V)
4941, 47, 48syl2anc 391 . 2 (𝜑 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V)
5040, 49syl5eqelr 2125 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  [wsb 1645  {cab 2026  ∀wral 2306  ∃wrex 2307  Vcvv 2557  {csn 3375  ⟨cop 3378  ∪ ciun 3657  {copab 3817   × cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910 This theorem is referenced by:  ovshftex  9420
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