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Theorem eloprabi 6101
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 2147 . . . . . 6 (𝑤 = 𝐴 → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
21anbi1d 461 . . . . 5 (𝑤 = 𝐴 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
323exbidv 1842 . . . 4 (𝑤 = 𝐴 → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
4 df-oprab 5785 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
53, 4elab2g 2834 . . 3 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
65ibi 175 . 2 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7 vex 2692 . . . . . . . . . . . 12 𝑥 ∈ V
8 vex 2692 . . . . . . . . . . . 12 𝑦 ∈ V
97, 8opex 4158 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
10 vex 2692 . . . . . . . . . . 11 𝑧 ∈ V
119, 10op1std 6053 . . . . . . . . . 10 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st𝐴) = ⟨𝑥, 𝑦⟩)
1211fveq2d 5432 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘(1st𝐴)) = (1st ‘⟨𝑥, 𝑦⟩))
137, 8op1st 6051 . . . . . . . . 9 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
1412, 13eqtr2di 2190 . . . . . . . 8 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑥 = (1st ‘(1st𝐴)))
15 eloprabi.1 . . . . . . . 8 (𝑥 = (1st ‘(1st𝐴)) → (𝜑𝜓))
1614, 15syl 14 . . . . . . 7 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑𝜓))
1711fveq2d 5432 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘(1st𝐴)) = (2nd ‘⟨𝑥, 𝑦⟩))
187, 8op2nd 6052 . . . . . . . . 9 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
1917, 18eqtr2di 2190 . . . . . . . 8 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑦 = (2nd ‘(1st𝐴)))
20 eloprabi.2 . . . . . . . 8 (𝑦 = (2nd ‘(1st𝐴)) → (𝜓𝜒))
2119, 20syl 14 . . . . . . 7 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜓𝜒))
229, 10op2ndd 6054 . . . . . . . . 9 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd𝐴) = 𝑧)
2322eqcomd 2146 . . . . . . . 8 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑧 = (2nd𝐴))
24 eloprabi.3 . . . . . . . 8 (𝑧 = (2nd𝐴) → (𝜒𝜃))
2523, 24syl 14 . . . . . . 7 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜒𝜃))
2616, 21, 253bitrd 213 . . . . . 6 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑𝜃))
2726biimpa 294 . . . . 5 ((𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
2827exlimiv 1578 . . . 4 (∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
2928exlimiv 1578 . . 3 (∃𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
3029exlimiv 1578 . 2 (∃𝑥𝑦𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
316, 30syl 14 1 (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wex 1469  wcel 1481  cop 3534  cfv 5130  {coprab 5782  1st c1st 6043  2nd c2nd 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fv 5138  df-oprab 5785  df-1st 6045  df-2nd 6046
This theorem is referenced by: (None)
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