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Theorem eloprabi 5745
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 2028 . . . . . 6 (w = A → (w = ⟨⟨x, y⟩, z⟩ ↔ A = ⟨⟨x, y⟩, z⟩))
21anbi1d 441 . . . . 5 (w = A → ((w = ⟨⟨x, y⟩, z φ) ↔ (A = ⟨⟨x, y⟩, z φ)))
323exbidv 1731 . . . 4 (w = A → (xyz(w = ⟨⟨x, y⟩, z φ) ↔ xyz(A = ⟨⟨x, y⟩, z φ)))
4 df-oprab 5440 . . . 4 {⟨⟨x, y⟩, z⟩ ∣ φ} = {wxyz(w = ⟨⟨x, y⟩, z φ)}
53, 4elab2g 2666 . . 3 (A {⟨⟨x, y⟩, z⟩ ∣ φ} → (A {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ xyz(A = ⟨⟨x, y⟩, z φ)))
65ibi 165 . 2 (A {⟨⟨x, y⟩, z⟩ ∣ φ} → xyz(A = ⟨⟨x, y⟩, z φ))
7 vex 2538 . . . . . . . . . . . 12 x V
8 vex 2538 . . . . . . . . . . . 12 y V
97, 8opex 3940 . . . . . . . . . . 11 x, y V
10 vex 2538 . . . . . . . . . . 11 z V
119, 10op1std 5698 . . . . . . . . . 10 (A = ⟨⟨x, y⟩, z⟩ → (1stA) = ⟨x, y⟩)
1211fveq2d 5107 . . . . . . . . 9 (A = ⟨⟨x, y⟩, z⟩ → (1st ‘(1stA)) = (1st ‘⟨x, y⟩))
137, 8op1st 5696 . . . . . . . . 9 (1st ‘⟨x, y⟩) = x
1412, 13syl6req 2071 . . . . . . . 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 5107 . . . . . . . . 9 (A = ⟨⟨x, y⟩, z⟩ → (2nd ‘(1stA)) = (2nd ‘⟨x, y⟩))
187, 8op2nd 5697 . . . . . . . . 9 (2nd ‘⟨x, y⟩) = y
1917, 18syl6req 2071 . . . . . . . 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 5699 . . . . . . . . 9 (A = ⟨⟨x, y⟩, z⟩ → (2ndA) = z)
2322eqcomd 2027 . . . . . . . 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 203 . . . . . 6 (A = ⟨⟨x, y⟩, z⟩ → (φθ))
2726biimpa 280 . . . . 5 ((A = ⟨⟨x, y⟩, z φ) → θ)
2827exlimiv 1471 . . . 4 (z(A = ⟨⟨x, y⟩, z φ) → θ)
2928exlimiv 1471 . . 3 (yz(A = ⟨⟨x, y⟩, z φ) → θ)
3029exlimiv 1471 . 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 97  wb 98   = wceq 1228  wex 1362   wcel 1374  cop 3353  cfv 4829  {coprab 5437  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-fv 4837  df-oprab 5440  df-1st 5690  df-2nd 5691
This theorem is referenced by: (None)
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