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Theorem opabex3rd 7989
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 1950 . . . . . . 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 1844 . . . . . . 7 (∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)) ↔ ∃𝑥(𝑦𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4 elxp 5711 . . . . . . . . . 10 (𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦})))
5 ancom 460 . . . . . . . . . . . 12 ((𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦}) ↔ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓}))
65anbi2i 623 . . . . . . . . . . 11 ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥𝜓} ∧ 𝑣 ∈ {𝑦})) ↔ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})))
762exbii 1845 . . . . . . . . . 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 645 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ (𝑣 ∈ {𝑦} ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
10 velsn 4646 . . . . . . . . . . . . . 14 (𝑣 ∈ {𝑦} ↔ 𝑣 = 𝑦)
1110anbi1i 624 . . . . . . . . . . . . 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 1844 . . . . . . . . . . 11 (∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ ∃𝑣(𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥𝜓})))
14 opeq2 4878 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑦⟩)
1514eqeq2d 2745 . . . . . . . . . . . . 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 2001 . . . . . . . . . . 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 1844 . . . . . . . . 9 (∃𝑤𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓}))
20 nfv 1911 . . . . . . . . . . 11 𝑥 𝑧 = ⟨𝑤, 𝑦
21 nfsab1 2719 . . . . . . . . . . 11 𝑥 𝑤 ∈ {𝑥𝜓}
2220, 21nfan 1896 . . . . . . . . . 10 𝑥(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥𝜓})
23 nfv 1911 . . . . . . . . . 10 𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
24 opeq1 4877 . . . . . . . . . . . 12 (𝑤 = 𝑥 → ⟨𝑤, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
2524eqeq2d 2745 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
26 df-clab 2712 . . . . . . . . . . . 12 (𝑤 ∈ {𝑥𝜓} ↔ [𝑤 / 𝑥]𝜓)
27 sbequ12 2248 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑥]𝜓))
2827equcoms 2016 . . . . . . . . . . . 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 2342 . . . . . . . . 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 623 . . . . . . 7 ((𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ (𝑦𝐴 ∧ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
341, 3, 333bitr4ri 304 . . . . . 6 ((𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
3534exbii 1844 . . . . 5 (∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})) ↔ ∃𝑦𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
36 excom 2159 . . . . 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 4999 . . . . 5 (𝑧 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ↔ ∃𝑦𝐴 𝑧 ∈ ({𝑥𝜓} × {𝑦}))
39 df-rex 3068 . . . . 5 (∃𝑦𝐴 𝑧 ∈ ({𝑥𝜓} × {𝑦}) ↔ ∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})))
4038, 39bitri 275 . . . 4 (𝑧 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ↔ ∃𝑦(𝑦𝐴𝑧 ∈ ({𝑥𝜓} × {𝑦})))
41 elopab 5536 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦𝐴𝜓)))
4237, 40, 413bitr4i 303 . . 3 (𝑧 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)})
4342eqriv 2731 . 2 𝑦𝐴 ({𝑥𝜓} × {𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)}
44 opabex3rd.1 . . 3 (𝜑𝐴𝑉)
45 opabex3rd.2 . . . . 5 ((𝜑𝑦𝐴) → {𝑥𝜓} ∈ V)
46 vsnex 5439 . . . . 5 {𝑦} ∈ V
47 xpexg 7768 . . . . 5 (({𝑥𝜓} ∈ V ∧ {𝑦} ∈ V) → ({𝑥𝜓} × {𝑦}) ∈ V)
4845, 46, 47sylancl 586 . . . 4 ((𝜑𝑦𝐴) → ({𝑥𝜓} × {𝑦}) ∈ V)
4948ralrimiva 3143 . . 3 (𝜑 → ∀𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V)
50 iunexg 7986 . . 3 ((𝐴𝑉 ∧ ∀𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V) → 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V)
5144, 49, 50syl2anc 584 . 2 (𝜑 𝑦𝐴 ({𝑥𝜓} × {𝑦}) ∈ V)
5243, 51eqeltrrid 2843 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  [wsb 2061  wcel 2105  {cab 2711  wral 3058  wrex 3067  Vcvv 3477  {csn 4630  cop 4636   ciun 4995  {copab 5209   × cxp 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-opab 5210  df-xp 5694  df-rel 5695
This theorem is referenced by:  satfvsuclem1  35343  satf0suclem  35359  fmlasuc0  35368
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