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Theorem eloprabi 7121
 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 2614 . . . . . 6 (𝑤 = 𝐴 → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
21anbi1d 737 . . . . 5 (𝑤 = 𝐴 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
323exbidv 1840 . . . 4 (𝑤 = 𝐴 → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
4 df-oprab 6553 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
53, 4elab2g 3322 . . 3 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
65ibi 255 . 2 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7 opex 4859 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
8 vex 3176 . . . . . . . . . . 11 𝑧 ∈ V
97, 8op1std 7069 . . . . . . . . . 10 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st𝐴) = ⟨𝑥, 𝑦⟩)
109fveq2d 6107 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘(1st𝐴)) = (1st ‘⟨𝑥, 𝑦⟩))
11 vex 3176 . . . . . . . . . 10 𝑥 ∈ V
12 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
1311, 12op1st 7067 . . . . . . . . 9 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
1410, 13syl6req 2661 . . . . . . . 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 6107 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘(1st𝐴)) = (2nd ‘⟨𝑥, 𝑦⟩))
1811, 12op2nd 7068 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
1917, 18syl6req 2661 . . . . . . . 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 7070 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd𝐴) = 𝑧)
2322eqcomd 2616 . . . . . . . 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 293 . . . . . 6 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑𝜃))
2726biimpa 500 . . . . 5 ((𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
2827exlimiv 1845 . . . 4 (∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
2928exlimiv 1845 . . 3 (∃𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
3029exlimiv 1845 . 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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ⟨cop 4131  ‘cfv 5804  {coprab 6550  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-oprab 6553  df-1st 7059  df-2nd 7060 This theorem is referenced by: (None)
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