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Theorem eloprabi 7277
Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
eloprabi.1 (𝑥 = (1st ‘(1st𝐴)) → (𝜑𝜓))
eloprabi.2 (𝑦 = (2nd ‘(1st𝐴)) → (𝜓𝜒))
eloprabi.3 (𝑧 = (2nd𝐴) → (𝜒𝜃))
Assertion
Ref Expression
eloprabi (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜃,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem eloprabi
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2655 . . . . . 6 (𝑤 = 𝐴 → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
21anbi1d 741 . . . . 5 (𝑤 = 𝐴 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
323exbidv 1893 . . . 4 (𝑤 = 𝐴 → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
4 df-oprab 6694 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
53, 4elab2g 3385 . . 3 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
65ibi 256 . 2 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7 opex 4962 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
8 vex 3234 . . . . . . . . . . 11 𝑧 ∈ V
97, 8op1std 7220 . . . . . . . . . 10 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st𝐴) = ⟨𝑥, 𝑦⟩)
109fveq2d 6233 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘(1st𝐴)) = (1st ‘⟨𝑥, 𝑦⟩))
11 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
12 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
1311, 12op1st 7218 . . . . . . . . 9 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
1410, 13syl6req 2702 . . . . . . . 8 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑥 = (1st ‘(1st𝐴)))
15 eloprabi.1 . . . . . . . 8 (𝑥 = (1st ‘(1st𝐴)) → (𝜑𝜓))
1614, 15syl 17 . . . . . . 7 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑𝜓))
179fveq2d 6233 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘(1st𝐴)) = (2nd ‘⟨𝑥, 𝑦⟩))
1811, 12op2nd 7219 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
1917, 18syl6req 2702 . . . . . . . 8 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑦 = (2nd ‘(1st𝐴)))
20 eloprabi.2 . . . . . . . 8 (𝑦 = (2nd ‘(1st𝐴)) → (𝜓𝜒))
2119, 20syl 17 . . . . . . 7 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜓𝜒))
227, 8op2ndd 7221 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd𝐴) = 𝑧)
2322eqcomd 2657 . . . . . . . 8 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑧 = (2nd𝐴))
24 eloprabi.3 . . . . . . . 8 (𝑧 = (2nd𝐴) → (𝜒𝜃))
2523, 24syl 17 . . . . . . 7 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜒𝜃))
2616, 21, 253bitrd 294 . . . . . 6 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑𝜃))
2726biimpa 500 . . . . 5 ((𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
2827exlimiv 1898 . . . 4 (∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
2928exlimiv 1898 . . 3 (∃𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
3029exlimiv 1898 . 2 (∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
316, 30syl 17 1 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  cop 4216  cfv 5926  {coprab 6691  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-oprab 6694  df-1st 7210  df-2nd 7211
This theorem is referenced by: (None)
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