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Theorem elopabi 5744
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 4391 . . . 4 Rel {⟨x, y⟩ ∣ φ}
2 1st2nd 5730 . . . 4 ((Rel {⟨x, y⟩ ∣ φ} A {⟨x, y⟩ ∣ φ}) → A = ⟨(1stA), (2ndA)⟩)
31, 2mpan 402 . . 3 (A {⟨x, y⟩ ∣ φ} → A = ⟨(1stA), (2ndA)⟩)
4 id 19 . . 3 (A {⟨x, y⟩ ∣ φ} → A {⟨x, y⟩ ∣ φ})
53, 4eqeltrrd 2097 . 2 (A {⟨x, y⟩ ∣ φ} → ⟨(1stA), (2ndA)⟩ {⟨x, y⟩ ∣ φ})
6 1stexg 5717 . . 3 (A {⟨x, y⟩ ∣ φ} → (1stA) V)
7 2ndexg 5718 . . 3 (A {⟨x, y⟩ ∣ φ} → (2ndA) V)
8 elopabi.1 . . . 4 (x = (1stA) → (φψ))
9 elopabi.2 . . . 4 (y = (2ndA) → (ψχ))
108, 9opelopabg 3979 . . 3 (((1stA) V (2ndA) V) → (⟨(1stA), (2ndA)⟩ {⟨x, y⟩ ∣ φ} ↔ χ))
116, 7, 10syl2anc 393 . 2 (A {⟨x, y⟩ ∣ φ} → (⟨(1stA), (2ndA)⟩ {⟨x, y⟩ ∣ φ} ↔ χ))
125, 11mpbid 135 1 (A {⟨x, y⟩ ∣ φ} → χ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  cop 3353  {copab 3791  Rel wrel 4277  cfv 4829  1st c1st 5688  2nd c2nd 5689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690  df-2nd 5691
This theorem is referenced by: (None)
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