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Theorem opabex3d 7311
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 2030 . . . . . 6 (∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2 an12 873 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)) ↔ (𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
32exbii 1923 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4 elxp 5288 . . . . . . . 8 (𝑧 ∈ ({𝑥} × {𝑦𝜓}) ↔ ∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})))
5 excom 2191 . . . . . . . . 9 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})))
6 an12 873 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
7 velsn 4337 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
87anbi1i 733 . . . . . . . . . . . . 13 ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
96, 8bitri 264 . . . . . . . . . . . 12 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
109exbii 1923 . . . . . . . . . . 11 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
11 vex 3343 . . . . . . . . . . . 12 𝑥 ∈ V
12 opeq1 4553 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
1312eqeq2d 2770 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
1413anbi1d 743 . . . . . . . . . . . 12 (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})))
1511, 14ceqsexv 3382 . . . . . . . . . . 11 (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
1610, 15bitri 264 . . . . . . . . . 10 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
1716exbii 1923 . . . . . . . . 9 (∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
185, 17bitri 264 . . . . . . . 8 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}))
19 nfv 1992 . . . . . . . . . 10 𝑦 𝑧 = ⟨𝑥, 𝑤
20 nfsab1 2750 . . . . . . . . . 10 𝑦 𝑤 ∈ {𝑦𝜓}
2119, 20nfan 1977 . . . . . . . . 9 𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓})
22 nfv 1992 . . . . . . . . 9 𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
23 opeq2 4554 . . . . . . . . . . 11 (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
2423eqeq2d 2770 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
25 sbequ12 2258 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
2625equcoms 2102 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
27 df-clab 2747 . . . . . . . . . . 11 (𝑤 ∈ {𝑦𝜓} ↔ [𝑤 / 𝑦]𝜓)
2826, 27syl6rbbr 279 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤 ∈ {𝑦𝜓} ↔ 𝜓))
2924, 28anbi12d 749 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3021, 22, 29cbvex 2417 . . . . . . . 8 (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
314, 18, 303bitri 286 . . . . . . 7 (𝑧 ∈ ({𝑥} × {𝑦𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
3231anbi2i 732 . . . . . 6 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
331, 3, 323bitr4ri 293 . . . . 5 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)))
3433exbii 1923 . . . 4 (∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)))
35 eliun 4676 . . . . 5 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜓}))
36 df-rex 3056 . . . . 5 (∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜓}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})))
3735, 36bitri 264 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜓})))
38 elopab 5133 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜓)))
3934, 37, 383bitr4i 292 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)})
4039eqriv 2757 . 2 𝑥𝐴 ({𝑥} × {𝑦𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)}
41 opabex3d.1 . . 3 (𝜑𝐴 ∈ V)
42 snex 5057 . . . . 5 {𝑥} ∈ V
43 opabex3d.2 . . . . 5 ((𝜑𝑥𝐴) → {𝑦𝜓} ∈ V)
44 xpexg 7126 . . . . 5 (({𝑥} ∈ V ∧ {𝑦𝜓} ∈ V) → ({𝑥} × {𝑦𝜓}) ∈ V)
4542, 43, 44sylancr 698 . . . 4 ((𝜑𝑥𝐴) → ({𝑥} × {𝑦𝜓}) ∈ V)
4645ralrimiva 3104 . . 3 (𝜑 → ∀𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V)
47 iunexg 7309 . . 3 ((𝐴 ∈ V ∧ ∀𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V) → 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V)
4841, 46, 47syl2anc 696 . 2 (𝜑 𝑥𝐴 ({𝑥} × {𝑦𝜓}) ∈ V)
4940, 48syl5eqelr 2844 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  [wsb 2046  wcel 2139  {cab 2746  wral 3050  wrex 3051  Vcvv 3340  {csn 4321  cop 4327   ciun 4672  {copab 4864   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057
This theorem is referenced by:  wksfval  26736  fpwrelmap  29838  cnvepresex  34446  opabresex0d  41830  upwlksfval  42244
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