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Theorem opabex3 7292
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 2032 . . . . . 6 (∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2 an12 616 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
32exbii 1923 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
4 elxp 5271 . . . . . . . 8 (𝑧 ∈ ({𝑥} × {𝑦𝜑}) ↔ ∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})))
5 excom 2197 . . . . . . . . 9 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})))
6 an12 616 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
7 velsn 4330 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
87anbi1i 602 . . . . . . . . . . . . 13 ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
96, 8bitri 264 . . . . . . . . . . . 12 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
109exbii 1923 . . . . . . . . . . 11 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
11 vex 3352 . . . . . . . . . . . 12 𝑥 ∈ V
12 opeq1 4537 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
1312eqeq2d 2780 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
1413anbi1d 607 . . . . . . . . . . . 12 (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
1511, 14ceqsexv 3391 . . . . . . . . . . 11 (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
1610, 15bitri 264 . . . . . . . . . 10 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
1716exbii 1923 . . . . . . . . 9 (∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
185, 17bitri 264 . . . . . . . 8 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
19 nfv 1994 . . . . . . . . . 10 𝑦 𝑧 = ⟨𝑥, 𝑤
20 nfsab1 2760 . . . . . . . . . 10 𝑦 𝑤 ∈ {𝑦𝜑}
2119, 20nfan 1979 . . . . . . . . 9 𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})
22 nfv 1994 . . . . . . . . 9 𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
23 opeq2 4538 . . . . . . . . . . 11 (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
2423eqeq2d 2780 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
25 sbequ12 2266 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
2625equcoms 2104 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
27 df-clab 2757 . . . . . . . . . . 11 (𝑤 ∈ {𝑦𝜑} ↔ [𝑤 / 𝑦]𝜑)
2826, 27syl6rbbr 279 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤 ∈ {𝑦𝜑} ↔ 𝜑))
2924, 28anbi12d 608 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3021, 22, 29cbvex 2432 . . . . . . . 8 (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
314, 18, 303bitri 286 . . . . . . 7 (𝑧 ∈ ({𝑥} × {𝑦𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3231anbi2i 601 . . . . . 6 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
331, 3, 323bitr4ri 293 . . . . 5 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)))
3433exbii 1923 . . . 4 (∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)))
35 eliun 4656 . . . . 5 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜑}))
36 df-rex 3066 . . . . 5 (∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜑}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})))
3735, 36bitri 264 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})))
38 elopab 5116 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)))
3934, 37, 383bitr4i 292 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
4039eqriv 2767 . 2 𝑥𝐴 ({𝑥} × {𝑦𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
41 opabex3.1 . . 3 𝐴 ∈ V
42 snex 5036 . . . . 5 {𝑥} ∈ V
43 opabex3.2 . . . . 5 (𝑥𝐴 → {𝑦𝜑} ∈ V)
44 xpexg 7106 . . . . 5 (({𝑥} ∈ V ∧ {𝑦𝜑} ∈ V) → ({𝑥} × {𝑦𝜑}) ∈ V)
4542, 43, 44sylancr 567 . . . 4 (𝑥𝐴 → ({𝑥} × {𝑦𝜑}) ∈ V)
4645rgen 3070 . . 3 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V
47 iunexg 7289 . . 3 ((𝐴 ∈ V ∧ ∀𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V) → 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V)
4841, 46, 47mp2an 664 . 2 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V
4940, 48eqeltrri 2846 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wex 1851  [wsb 2048  wcel 2144  {cab 2756  wral 3060  wrex 3061  Vcvv 3349  {csn 4314  cop 4320   ciun 4652  {copab 4844   × cxp 5247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  dvdsrval  18852  eulerpartlemgvv  30772  cnfinltrel  33571
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