ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elopabi Structured version   GIF version

Theorem elopabi 5763
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 4407 . . . 4 Rel {⟨x, y⟩ ∣ φ}
2 1st2nd 5749 . . . 4 ((Rel {⟨x, y⟩ ∣ φ} A {⟨x, y⟩ ∣ φ}) → A = ⟨(1stA), (2ndA)⟩)
31, 2mpan 400 . . 3 (A {⟨x, y⟩ ∣ φ} → A = ⟨(1stA), (2ndA)⟩)
4 id 19 . . 3 (A {⟨x, y⟩ ∣ φ} → A {⟨x, y⟩ ∣ φ})
53, 4eqeltrrd 2112 . 2 (A {⟨x, y⟩ ∣ φ} → ⟨(1stA), (2ndA)⟩ {⟨x, y⟩ ∣ φ})
6 1stexg 5736 . . 3 (A {⟨x, y⟩ ∣ φ} → (1stA) V)
7 2ndexg 5737 . . 3 (A {⟨x, y⟩ ∣ φ} → (2ndA) V)
8 elopabi.1 . . . 4 (x = (1stA) → (φψ))
9 elopabi.2 . . . 4 (y = (2ndA) → (ψχ))
108, 9opelopabg 3996 . . 3 (((1stA) V (2ndA) V) → (⟨(1stA), (2ndA)⟩ {⟨x, y⟩ ∣ φ} ↔ χ))
116, 7, 10syl2anc 391 . 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 1242   wcel 1390  Vcvv 2551  cop 3370  {copab 3808  Rel wrel 4293  cfv 4845  1st c1st 5707  2nd c2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator