MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eloprabi Structured version   Visualization version   GIF version

Theorem eloprabi 7433
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 2769 . . . . . 6 (𝑤 = 𝐴 → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
21anbi1d 623 . . . . 5 (𝑤 = 𝐴 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
323exbidv 2020 . . . 4 (𝑤 = 𝐴 → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
4 df-oprab 6846 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
53, 4elab2g 3508 . . 3 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
65ibi 258 . 2 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7 opex 5088 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
8 vex 3353 . . . . . . . . . . 11 𝑧 ∈ V
97, 8op1std 7376 . . . . . . . . . 10 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st𝐴) = ⟨𝑥, 𝑦⟩)
109fveq2d 6379 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘(1st𝐴)) = (1st ‘⟨𝑥, 𝑦⟩))
11 vex 3353 . . . . . . . . . 10 𝑥 ∈ V
12 vex 3353 . . . . . . . . . 10 𝑦 ∈ V
1311, 12op1st 7374 . . . . . . . . 9 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
1410, 13syl6req 2816 . . . . . . . 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 6379 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘(1st𝐴)) = (2nd ‘⟨𝑥, 𝑦⟩))
1811, 12op2nd 7375 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
1917, 18syl6req 2816 . . . . . . . 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 7377 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd𝐴) = 𝑧)
2322eqcomd 2771 . . . . . . . 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 296 . . . . . 6 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑𝜃))
2726biimpa 468 . . . . 5 ((𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
2827exlimiv 2025 . . . 4 (∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
2928exlimiv 2025 . . 3 (∃𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
3029exlimiv 2025 . 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 197  wa 384   = wceq 1652  wex 1874  wcel 2155  cop 4340  cfv 6068  {coprab 6843  1st c1st 7364  2nd c2nd 7365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fv 6076  df-oprab 6846  df-1st 7366  df-2nd 7367
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator