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Theorem opabex3rd 7705
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 1961 . . . . . . 7 (∃𝑥(𝑦𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (𝑦𝐴 ∧ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2 an12 645 . . . . . . . 8 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)) ↔ (𝑦𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
32exbii 1854 . . . . . . 7 (∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)) ↔ ∃𝑥(𝑦𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4 elxp 5558 . . . . . . . . . 10 (𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦})))
5 ancom 464 . . . . . . . . . . . 12 ((𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦}) ↔ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓}))
65anbi2i 626 . . . . . . . . . . 11 ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦})) ↔ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})))
762exbii 1855 . . . . . . . . . 10 (∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦})) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})))
84, 7bitri 278 . . . . . . . . 9 (𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})))
9 an12 645 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑣 ∈ {𝑦} ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
10 velsn 4542 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑦} ↔ 𝑣 = 𝑦)
1110anbi1i 627 . . . . . . . . . . . . 13 ((𝑣 ∈ {𝑦} ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
129, 11bitri 278 . . . . . . . . . . . 12 ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
1312exbii 1854 . . . . . . . . . . 11 (∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ ∃𝑣(𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
14 opeq2 4771 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑦⟩)
1514eqeq2d 2750 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩))
1615anbi1d 633 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓}) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
1716equsexvw 2016 . . . . . . . . . . 11 (∃𝑣(𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}))
1813, 17bitri 278 . . . . . . . . . 10 (∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}))
1918exbii 1854 . . . . . . . . 9 (∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}))
20 nfv 1921 . . . . . . . . . . 11 𝑥 𝑧 = ⟨𝑤, 𝑦
21 nfsab1 2726 . . . . . . . . . . 11 𝑥 𝑤 ∈ {𝑥𝜓}
2220, 21nfan 1906 . . . . . . . . . 10 𝑥(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓})
23 nfv 1921 . . . . . . . . . 10 𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
24 opeq1 4769 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ⟨𝑤, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
2524eqeq2d 2750 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
26 df-clab 2718 . . . . . . . . . . . 12 (𝑤 ∈ {𝑥𝜓} ↔ [𝑤 / 𝑥]𝜓)
27 sbequ12 2253 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑥]𝜓))
2827equcoms 2032 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝜓 ↔ [𝑤 / 𝑥]𝜓))
2926, 28bitr4id 293 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑤 ∈ {𝑥𝜓} ↔ 𝜓))
3025, 29anbi12d 634 . . . . . . . . . 10 (𝑤 = 𝑥 → ((𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3122, 23, 30cbvexv1 2345 . . . . . . . . 9 (∃𝑤(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
328, 19, 313bitri 300 . . . . . . . 8 (𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
3332anbi2i 626 . . . . . . 7 ((𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ (𝑦𝐴 ∧ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
341, 3, 333bitr4ri 307 . . . . . 6 ((𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
3534exbii 1854 . . . . 5 (∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ ∃𝑦𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
36 excom 2170 . . . . 5 (∃𝑦𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
3735, 36bitri 278 . . . 4 (∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
38 eliun 4895 . . . . 5 (𝑧 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ↔ ∃𝑦𝐴 𝑧 ∈ ({𝑥𝜓} × {𝑦}))
39 df-rex 3060 . . . . 5 (∃𝑦𝐴 𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})))
4038, 39bitri 278 . . . 4 (𝑧 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ↔ ∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})))
41 elopab 5392 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
4237, 40, 413bitr4i 306 . . 3 (𝑧 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)})
4342eqriv 2736 . 2 𝑦𝐴 ({𝑥𝜓} × {𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)}
44 opabex3rd.1 . . 3 (𝜑𝐴𝑉)
45 opabex3rd.2 . . . . 5 ((𝜑𝑦𝐴) → {𝑥𝜓} ∈ V)
46 snex 5308 . . . . 5 {𝑦} ∈ V
47 xpexg 7504 . . . . 5 (({𝑥𝜓} ∈ V ∧ {𝑦} ∈ V) → ({𝑥𝜓} × {𝑦}) ∈ V)
4845, 46, 47sylancl 589 . . . 4 ((𝜑𝑦𝐴) → ({𝑥𝜓} × {𝑦}) ∈ V)
4948ralrimiva 3097 . . 3 (𝜑 → ∀𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V)
50 iunexg 7702 . . 3 ((𝐴𝑉 ∧ ∀𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V) → 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V)
5144, 49, 50syl2anc 587 . 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 209  wa 399   = wceq 1542  wex 1786  [wsb 2074  wcel 2114  {cab 2717  wral 3054  wrex 3055  Vcvv 3400  {csn 4526  cop 4532   ciun 4891  {copab 5102   × cxp 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358
This theorem is referenced by:  satfvsuclem1  32905  satf0suclem  32921  fmlasuc0  32930
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