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Theorem opabex3 5931
 Description: Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
opabex3.1 𝐴 ∈ V
opabex3.2 (𝑥𝐴 → {𝑦𝜑} ∈ V)
Assertion
Ref Expression
opabex3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabex3
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.42v 1841 . . . . . 6 (∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2 an12 529 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
32exbii 1548 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
4 elxp 4484 . . . . . . . 8 (𝑧 ∈ ({𝑥} × {𝑦𝜑}) ↔ ∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})))
5 excom 1606 . . . . . . . . 9 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})))
6 an12 529 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
7 velsn 3483 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
87anbi1i 447 . . . . . . . . . . . . 13 ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
96, 8bitri 183 . . . . . . . . . . . 12 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
109exbii 1548 . . . . . . . . . . 11 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
11 vex 2636 . . . . . . . . . . . 12 𝑥 ∈ V
12 opeq1 3644 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
1312eqeq2d 2106 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
1413anbi1d 454 . . . . . . . . . . . 12 (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
1511, 14ceqsexv 2672 . . . . . . . . . . 11 (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
1610, 15bitri 183 . . . . . . . . . 10 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
1716exbii 1548 . . . . . . . . 9 (∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
185, 17bitri 183 . . . . . . . 8 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
19 nfv 1473 . . . . . . . . . 10 𝑦 𝑧 = ⟨𝑥, 𝑤
20 nfsab1 2085 . . . . . . . . . 10 𝑦 𝑤 ∈ {𝑦𝜑}
2119, 20nfan 1509 . . . . . . . . 9 𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})
22 nfv 1473 . . . . . . . . 9 𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
23 opeq2 3645 . . . . . . . . . . 11 (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
2423eqeq2d 2106 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
25 sbequ12 1708 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
2625equcoms 1648 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
27 df-clab 2082 . . . . . . . . . . 11 (𝑤 ∈ {𝑦𝜑} ↔ [𝑤 / 𝑦]𝜑)
2826, 27syl6rbbr 198 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤 ∈ {𝑦𝜑} ↔ 𝜑))
2924, 28anbi12d 458 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3021, 22, 29cbvex 1693 . . . . . . . 8 (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
314, 18, 303bitri 205 . . . . . . 7 (𝑧 ∈ ({𝑥} × {𝑦𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3231anbi2i 446 . . . . . 6 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
331, 3, 323bitr4ri 212 . . . . 5 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)))
3433exbii 1548 . . . 4 (∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)))
35 eliun 3756 . . . . 5 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜑}))
36 df-rex 2376 . . . . 5 (∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜑}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})))
3735, 36bitri 183 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})))
38 elopab 4109 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)))
3934, 37, 383bitr4i 211 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
4039eqriv 2092 . 2 𝑥𝐴 ({𝑥} × {𝑦𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
41 opabex3.1 . . 3 𝐴 ∈ V
42 snexg 4040 . . . . . 6 (𝑥 ∈ V → {𝑥} ∈ V)
4311, 42ax-mp 7 . . . . 5 {𝑥} ∈ V
44 opabex3.2 . . . . 5 (𝑥𝐴 → {𝑦𝜑} ∈ V)
45 xpexg 4581 . . . . 5 (({𝑥} ∈ V ∧ {𝑦𝜑} ∈ V) → ({𝑥} × {𝑦𝜑}) ∈ V)
4643, 44, 45sylancr 406 . . . 4 (𝑥𝐴 → ({𝑥} × {𝑦𝜑}) ∈ V)
4746rgen 2439 . . 3 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V
48 iunexg 5928 . . 3 ((𝐴 ∈ V ∧ ∀𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V) → 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V)
4941, 47, 48mp2an 418 . 2 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V
5040, 49eqeltrri 2168 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296  ∃wex 1433   ∈ wcel 1445  [wsb 1699  {cab 2081  ∀wral 2370  ∃wrex 2371  Vcvv 2633  {csn 3466  ⟨cop 3469  ∪ ciun 3752  {copab 3920   × cxp 4465 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057 This theorem is referenced by: (None)
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