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Theorem elopabi 6086
Description: A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
Hypotheses
Ref Expression
elopabi.1 (𝑥 = (1st𝐴) → (𝜑𝜓))
elopabi.2 (𝑦 = (2nd𝐴) → (𝜓𝜒))
Assertion
Ref Expression
elopabi (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem elopabi
StepHypRef Expression
1 relopab 4661 . . . 4 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 1st2nd 6072 . . . 4 ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
31, 2mpan 420 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
4 id 19 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
53, 4eqeltrrd 2215 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6 1stexg 6058 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (1st𝐴) ∈ V)
7 2ndexg 6059 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (2nd𝐴) ∈ V)
8 elopabi.1 . . . 4 (𝑥 = (1st𝐴) → (𝜑𝜓))
9 elopabi.2 . . . 4 (𝑦 = (2nd𝐴) → (𝜓𝜒))
108, 9opelopabg 4185 . . 3 (((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V) → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
116, 7, 10syl2anc 408 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
125, 11mpbid 146 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  Vcvv 2681  cop 3525  {copab 3983  Rel wrel 4539  cfv 5118  1st c1st 6029  2nd c2nd 6030
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-1st 6031  df-2nd 6032
This theorem is referenced by:  aprcl  8401
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