Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  opabex3 Structured version   Visualization version   GIF version

Theorem opabex3 7652
 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 1954 . . . . . 6 (∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2 an12 644 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
32exbii 1849 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
4 elxp 5542 . . . . . . . 8 (𝑧 ∈ ({𝑥} × {𝑦𝜑}) ↔ ∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})))
5 excom 2166 . . . . . . . . 9 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})))
6 an12 644 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
7 velsn 4541 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
87anbi1i 626 . . . . . . . . . . . . 13 ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
96, 8bitri 278 . . . . . . . . . . . 12 ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
109exbii 1849 . . . . . . . . . . 11 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
11 opeq1 4763 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
1211eqeq2d 2809 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
1312anbi1d 632 . . . . . . . . . . . 12 (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})))
1413equsexvw 2011 . . . . . . . . . . 11 (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
1510, 14bitri 278 . . . . . . . . . 10 (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
1615exbii 1849 . . . . . . . . 9 (∃𝑤𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
175, 16bitri 278 . . . . . . . 8 (∃𝑣𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}))
18 nfv 1915 . . . . . . . . . 10 𝑦 𝑧 = ⟨𝑥, 𝑤
19 nfsab1 2785 . . . . . . . . . 10 𝑦 𝑤 ∈ {𝑦𝜑}
2018, 19nfan 1900 . . . . . . . . 9 𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑})
21 nfv 1915 . . . . . . . . 9 𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
22 opeq2 4765 . . . . . . . . . . 11 (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
2322eqeq2d 2809 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
24 df-clab 2777 . . . . . . . . . . 11 (𝑤 ∈ {𝑦𝜑} ↔ [𝑤 / 𝑦]𝜑)
25 sbequ12 2250 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
2625equcoms 2027 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
2724, 26bitr4id 293 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤 ∈ {𝑦𝜑} ↔ 𝜑))
2823, 27anbi12d 633 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2920, 21, 28cbvexv1 2351 . . . . . . . 8 (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
304, 17, 293bitri 300 . . . . . . 7 (𝑧 ∈ ({𝑥} × {𝑦𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3130anbi2i 625 . . . . . 6 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
321, 3, 313bitr4ri 307 . . . . 5 ((𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)))
3332exbii 1849 . . . 4 (∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)))
34 eliun 4885 . . . . 5 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ↔ ∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜑}))
35 df-rex 3112 . . . . 5 (∃𝑥𝐴 𝑧 ∈ ({𝑥} × {𝑦𝜑}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})))
3634, 35bitri 278 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ↔ ∃𝑥(𝑥𝐴𝑧 ∈ ({𝑥} × {𝑦𝜑})))
37 elopab 5379 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝜑)))
3833, 36, 373bitr4i 306 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
3938eqriv 2795 . 2 𝑥𝐴 ({𝑥} × {𝑦𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
40 opabex3.1 . . 3 𝐴 ∈ V
41 snex 5297 . . . . 5 {𝑥} ∈ V
42 opabex3.2 . . . . 5 (𝑥𝐴 → {𝑦𝜑} ∈ V)
43 xpexg 7455 . . . . 5 (({𝑥} ∈ V ∧ {𝑦𝜑} ∈ V) → ({𝑥} × {𝑦𝜑}) ∈ V)
4441, 42, 43sylancr 590 . . . 4 (𝑥𝐴 → ({𝑥} × {𝑦𝜑}) ∈ V)
4544rgen 3116 . . 3 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V
46 iunexg 7648 . . 3 ((𝐴 ∈ V ∧ ∀𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V) → 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V)
4740, 45, 46mp2an 691 . 2 𝑥𝐴 ({𝑥} × {𝑦𝜑}) ∈ V
4839, 47eqeltrri 2887 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781  [wsb 2069   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  Vcvv 3441  {csn 4525  ⟨cop 4531  ∪ ciun 4881  {copab 5092   × cxp 5517 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332 This theorem is referenced by:  dvdsrval  19394  eulerpartlemgvv  31756
 Copyright terms: Public domain W3C validator