MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opabex3rd Structured version   Visualization version   GIF version

Theorem opabex3rd 7900
Description: Existence of an ordered pair abstraction if the second components are elements of a set. (Contributed by AV, 17-Sep-2023.) (Revised by AV, 9-Aug-2024.)
Hypotheses
Ref Expression
opabex3rd.1 (𝜑𝐴𝑉)
opabex3rd.2 ((𝜑𝑦𝐴) → {𝑥𝜓} ∈ V)
Assertion
Ref Expression
opabex3rd (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)} ∈ V)
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem opabex3rd
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.42v 1958 . . . . . . 7 (∃𝑥(𝑦𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (𝑦𝐴 ∧ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2 an12 644 . . . . . . . 8 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)) ↔ (𝑦𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
32exbii 1851 . . . . . . 7 (∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)) ↔ ∃𝑥(𝑦𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4 elxp 5657 . . . . . . . . . 10 (𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦})))
5 ancom 462 . . . . . . . . . . . 12 ((𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦}) ↔ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓}))
65anbi2i 624 . . . . . . . . . . 11 ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦})) ↔ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})))
762exbii 1852 . . . . . . . . . 10 (∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦})) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})))
84, 7bitri 275 . . . . . . . . 9 (𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})))
9 an12 644 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑣 ∈ {𝑦} ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
10 velsn 4603 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑦} ↔ 𝑣 = 𝑦)
1110anbi1i 625 . . . . . . . . . . . . 13 ((𝑣 ∈ {𝑦} ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
129, 11bitri 275 . . . . . . . . . . . 12 ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
1312exbii 1851 . . . . . . . . . . 11 (∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ ∃𝑣(𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
14 opeq2 4832 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑦⟩)
1514eqeq2d 2744 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩))
1615anbi1d 631 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓}) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
1716equsexvw 2009 . . . . . . . . . . 11 (∃𝑣(𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}))
1813, 17bitri 275 . . . . . . . . . 10 (∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}))
1918exbii 1851 . . . . . . . . 9 (∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}))
20 nfv 1918 . . . . . . . . . . 11 𝑥 𝑧 = ⟨𝑤, 𝑦
21 nfsab1 2718 . . . . . . . . . . 11 𝑥 𝑤 ∈ {𝑥𝜓}
2220, 21nfan 1903 . . . . . . . . . 10 𝑥(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓})
23 nfv 1918 . . . . . . . . . 10 𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
24 opeq1 4831 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ⟨𝑤, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
2524eqeq2d 2744 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
26 df-clab 2711 . . . . . . . . . . . 12 (𝑤 ∈ {𝑥𝜓} ↔ [𝑤 / 𝑥]𝜓)
27 sbequ12 2244 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑥]𝜓))
2827equcoms 2024 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝜓 ↔ [𝑤 / 𝑥]𝜓))
2926, 28bitr4id 290 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑤 ∈ {𝑥𝜓} ↔ 𝜓))
3025, 29anbi12d 632 . . . . . . . . . 10 (𝑤 = 𝑥 → ((𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3122, 23, 30cbvexv1 2339 . . . . . . . . 9 (∃𝑤(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
328, 19, 313bitri 297 . . . . . . . 8 (𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
3332anbi2i 624 . . . . . . 7 ((𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ (𝑦𝐴 ∧ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
341, 3, 333bitr4ri 304 . . . . . 6 ((𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
3534exbii 1851 . . . . 5 (∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ ∃𝑦𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
36 excom 2163 . . . . 5 (∃𝑦𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
3735, 36bitri 275 . . . 4 (∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
38 eliun 4959 . . . . 5 (𝑧 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ↔ ∃𝑦𝐴 𝑧 ∈ ({𝑥𝜓} × {𝑦}))
39 df-rex 3071 . . . . 5 (∃𝑦𝐴 𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})))
4038, 39bitri 275 . . . 4 (𝑧 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ↔ ∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})))
41 elopab 5485 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
4237, 40, 413bitr4i 303 . . 3 (𝑧 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)})
4342eqriv 2730 . 2 𝑦𝐴 ({𝑥𝜓} × {𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)}
44 opabex3rd.1 . . 3 (𝜑𝐴𝑉)
45 opabex3rd.2 . . . . 5 ((𝜑𝑦𝐴) → {𝑥𝜓} ∈ V)
46 vsnex 5387 . . . . 5 {𝑦} ∈ V
47 xpexg 7685 . . . . 5 (({𝑥𝜓} ∈ V ∧ {𝑦} ∈ V) → ({𝑥𝜓} × {𝑦}) ∈ V)
4845, 46, 47sylancl 587 . . . 4 ((𝜑𝑦𝐴) → ({𝑥𝜓} × {𝑦}) ∈ V)
4948ralrimiva 3140 . . 3 (𝜑 → ∀𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V)
50 iunexg 7897 . . 3 ((𝐴𝑉 ∧ ∀𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V) → 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V)
5144, 49, 50syl2anc 585 . 2 (𝜑 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V)
5243, 51eqeltrrid 2839 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  [wsb 2068  wcel 2107  {cab 2710  wral 3061  wrex 3070  Vcvv 3444  {csn 4587  cop 4593   ciun 4955  {copab 5168   × cxp 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-opab 5169  df-xp 5640  df-rel 5641
This theorem is referenced by:  satfvsuclem1  34010  satf0suclem  34026  fmlasuc0  34035
  Copyright terms: Public domain W3C validator