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Theorem dfoprab3s 5758
 Description: A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfoprab3s {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ (w (V × V) [(1stw) / x][(2ndw) / y]φ)}
Distinct variable groups:   φ,w   x,y,z,w
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem dfoprab3s
StepHypRef Expression
1 dfoprab2 5494 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
2 nfsbc1v 2776 . . . . 5 x[(1stw) / x][(2ndw) / y]φ
3219.41 1573 . . . 4 (x(y w = ⟨x, y [(1stw) / x][(2ndw) / y]φ) ↔ (xy w = ⟨x, y [(1stw) / x][(2ndw) / y]φ))
4 sbcopeq1a 5755 . . . . . . . 8 (w = ⟨x, y⟩ → ([(1stw) / x][(2ndw) / y]φφ))
54pm5.32i 427 . . . . . . 7 ((w = ⟨x, y [(1stw) / x][(2ndw) / y]φ) ↔ (w = ⟨x, y φ))
65exbii 1493 . . . . . 6 (y(w = ⟨x, y [(1stw) / x][(2ndw) / y]φ) ↔ y(w = ⟨x, y φ))
7 nfcv 2175 . . . . . . . 8 y(1stw)
8 nfsbc1v 2776 . . . . . . . 8 y[(2ndw) / y]φ
97, 8nfsbc 2778 . . . . . . 7 y[(1stw) / x][(2ndw) / y]φ
10919.41 1573 . . . . . 6 (y(w = ⟨x, y [(1stw) / x][(2ndw) / y]φ) ↔ (y w = ⟨x, y [(1stw) / x][(2ndw) / y]φ))
116, 10bitr3i 175 . . . . 5 (y(w = ⟨x, y φ) ↔ (y w = ⟨x, y [(1stw) / x][(2ndw) / y]φ))
1211exbii 1493 . . . 4 (xy(w = ⟨x, y φ) ↔ x(y w = ⟨x, y [(1stw) / x][(2ndw) / y]φ))
13 elvv 4345 . . . . 5 (w (V × V) ↔ xy w = ⟨x, y⟩)
1413anbi1i 431 . . . 4 ((w (V × V) [(1stw) / x][(2ndw) / y]φ) ↔ (xy w = ⟨x, y [(1stw) / x][(2ndw) / y]φ))
153, 12, 143bitr4i 201 . . 3 (xy(w = ⟨x, y φ) ↔ (w (V × V) [(1stw) / x][(2ndw) / y]φ))
1615opabbii 3815 . 2 {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} = {⟨w, z⟩ ∣ (w (V × V) [(1stw) / x][(2ndw) / y]φ)}
171, 16eqtri 2057 1 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ (w (V × V) [(1stw) / x][(2ndw) / y]φ)}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  [wsbc 2758  ⟨cop 3370  {copab 3808   × cxp 4286  ‘cfv 4845  {coprab 5456  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-fv 4853  df-oprab 5459  df-1st 5709  df-2nd 5710 This theorem is referenced by:  dfoprab3  5759
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