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Theorem dfopab2 5757
Description: A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfopab2 {⟨x, y⟩ ∣ φ} = {z (V × V) ∣ [(1stz) / x][(2ndz) / y]φ}
Distinct variable groups:   φ,z   x,y,z
Allowed substitution hints:   φ(x,y)

Proof of Theorem dfopab2
StepHypRef Expression
1 nfsbc1v 2776 . . . . 5 x[(1stz) / x][(2ndz) / y]φ
2119.41 1573 . . . 4 (x(y z = ⟨x, y [(1stz) / x][(2ndz) / y]φ) ↔ (xy z = ⟨x, y [(1stz) / x][(2ndz) / y]φ))
3 sbcopeq1a 5755 . . . . . . . 8 (z = ⟨x, y⟩ → ([(1stz) / x][(2ndz) / y]φφ))
43pm5.32i 427 . . . . . . 7 ((z = ⟨x, y [(1stz) / x][(2ndz) / y]φ) ↔ (z = ⟨x, y φ))
54exbii 1493 . . . . . 6 (y(z = ⟨x, y [(1stz) / x][(2ndz) / y]φ) ↔ y(z = ⟨x, y φ))
6 nfcv 2175 . . . . . . . 8 y(1stz)
7 nfsbc1v 2776 . . . . . . . 8 y[(2ndz) / y]φ
86, 7nfsbc 2778 . . . . . . 7 y[(1stz) / x][(2ndz) / y]φ
9819.41 1573 . . . . . 6 (y(z = ⟨x, y [(1stz) / x][(2ndz) / y]φ) ↔ (y z = ⟨x, y [(1stz) / x][(2ndz) / y]φ))
105, 9bitr3i 175 . . . . 5 (y(z = ⟨x, y φ) ↔ (y z = ⟨x, y [(1stz) / x][(2ndz) / y]φ))
1110exbii 1493 . . . 4 (xy(z = ⟨x, y φ) ↔ x(y z = ⟨x, y [(1stz) / x][(2ndz) / y]φ))
12 elvv 4345 . . . . 5 (z (V × V) ↔ xy z = ⟨x, y⟩)
1312anbi1i 431 . . . 4 ((z (V × V) [(1stz) / x][(2ndz) / y]φ) ↔ (xy z = ⟨x, y [(1stz) / x][(2ndz) / y]φ))
142, 11, 133bitr4i 201 . . 3 (xy(z = ⟨x, y φ) ↔ (z (V × V) [(1stz) / x][(2ndz) / y]φ))
1514abbii 2150 . 2 {zxy(z = ⟨x, y φ)} = {z ∣ (z (V × V) [(1stz) / x][(2ndz) / y]φ)}
16 df-opab 3810 . 2 {⟨x, y⟩ ∣ φ} = {zxy(z = ⟨x, y φ)}
17 df-rab 2309 . 2 {z (V × V) ∣ [(1stz) / x][(2ndz) / y]φ} = {z ∣ (z (V × V) [(1stz) / x][(2ndz) / y]φ)}
1815, 16, 173eqtr4i 2067 1 {⟨x, y⟩ ∣ φ} = {z (V × V) ∣ [(1stz) / x][(2ndz) / y]φ}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  {crab 2304  Vcvv 2551  [wsbc 2758  cop 3370  {copab 3808   × cxp 4286  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-rab 2309  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-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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