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Theorem oprabv 6868
Description: If a pair and a class are in a relationship given by a class abstraction of a collection of nested ordered pairs, the involved classes are sets. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
oprabv (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))
Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem oprabv
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 reloprab 6867 . . 3 Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
2 brrelex12 5312 . . 3 ((Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∧ ⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍) → (⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V))
31, 2mpan 708 . 2 (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V))
4 df-br 4805 . . . . 5 (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 ↔ ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
5 opex 5081 . . . . . . . . 9 𝑋, 𝑌⟩ ∈ V
6 nfcv 2902 . . . . . . . . . . . . . 14 𝑤𝑋, 𝑌
76nfeq1 2916 . . . . . . . . . . . . 13 𝑤𝑋, 𝑌⟩ = ⟨𝑥, 𝑦
8 nfv 1992 . . . . . . . . . . . . 13 𝑤𝜑
97, 8nfan 1977 . . . . . . . . . . . 12 𝑤(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
109nfex 2301 . . . . . . . . . . 11 𝑤𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1110nfex 2301 . . . . . . . . . 10 𝑤𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
12 nfcv 2902 . . . . . . . . . . . . . 14 𝑧𝑋, 𝑌
1312nfeq1 2916 . . . . . . . . . . . . 13 𝑧𝑋, 𝑌⟩ = ⟨𝑥, 𝑦
14 nfsbc1v 3596 . . . . . . . . . . . . 13 𝑧[𝑍 / 𝑧]𝜑
1513, 14nfan 1977 . . . . . . . . . . . 12 𝑧(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)
1615nfex 2301 . . . . . . . . . . 11 𝑧𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)
1716nfex 2301 . . . . . . . . . 10 𝑧𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)
18 eqeq1 2764 . . . . . . . . . . . 12 (𝑤 = ⟨𝑋, 𝑌⟩ → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩))
1918anbi1d 743 . . . . . . . . . . 11 (𝑤 = ⟨𝑋, 𝑌⟩ → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
20192exbidv 2001 . . . . . . . . . 10 (𝑤 = ⟨𝑋, 𝑌⟩ → (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
21 sbceq1a 3587 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝜑[𝑍 / 𝑧]𝜑))
2221anbi2d 742 . . . . . . . . . . 11 (𝑧 = 𝑍 → ((⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)))
23222exbidv 2001 . . . . . . . . . 10 (𝑧 = 𝑍 → (∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)))
2411, 17, 20, 23opelopabgf 5145 . . . . . . . . 9 ((⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V) → (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)))
255, 24mpan 708 . . . . . . . 8 (𝑍 ∈ V → (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)))
26 eqcom 2767 . . . . . . . . . . . . . . 15 (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
27 vex 3343 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
28 vex 3343 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
2927, 28opth 5093 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
3026, 29bitri 264 . . . . . . . . . . . . . 14 (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
31 eqvisset 3351 . . . . . . . . . . . . . . 15 (𝑥 = 𝑋𝑋 ∈ V)
32 eqvisset 3351 . . . . . . . . . . . . . . 15 (𝑦 = 𝑌𝑌 ∈ V)
3331, 32anim12i 591 . . . . . . . . . . . . . 14 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3430, 33sylbi 207 . . . . . . . . . . . . 13 (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3534adantr 472 . . . . . . . . . . . 12 ((⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3635exlimivv 2009 . . . . . . . . . . 11 (∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3736anim1i 593 . . . . . . . . . 10 ((∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) ∧ 𝑍 ∈ V) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ V))
38 df-3an 1074 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ V))
3937, 38sylibr 224 . . . . . . . . 9 ((∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) ∧ 𝑍 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))
4039expcom 450 . . . . . . . 8 (𝑍 ∈ V → (∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
4125, 40sylbid 230 . . . . . . 7 (𝑍 ∈ V → (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
4241com12 32 . . . . . 6 (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} → (𝑍 ∈ V → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
43 dfoprab2 6866 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4442, 43eleq2s 2857 . . . . 5 (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝑍 ∈ V → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
454, 44sylbi 207 . . . 4 (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑍 ∈ V → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
4645com12 32 . . 3 (𝑍 ∈ V → (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
4746adantl 473 . 2 ((⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V) → (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
483, 47mpcom 38 1 (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  [wsbc 3576  cop 4327   class class class wbr 4804  {copab 4864  Rel wrel 5271  {coprab 6814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-oprab 6817
This theorem is referenced by: (None)
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