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) ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ)}

Distinct variable groups:   φ,w   x,y,z,w

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem dfoprab3s

Step	Hyp	Ref	Expression
1		dfoprab2 5494	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
2		nfsbc1v 2776	. . . . 5 ⊢ Ⅎx[(1st ‘w) / x][(2nd ‘w) / y]φ
3	2	19.41 1573	. . . 4 ⊢ (∃x(∃y w = ⟨x, y⟩ ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ) ↔ (∃x∃y w = ⟨x, y⟩ ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ))
4		sbcopeq1a 5755	. . . . . . . 8 ⊢ (w = ⟨x, y⟩ → ([(1st ‘w) / x][(2nd ‘w) / y]φ ↔ φ))
5	4	pm5.32i 427	. . . . . . 7 ⊢ ((w = ⟨x, y⟩ ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ) ↔ (w = ⟨x, y⟩ ∧ φ))
6	5	exbii 1493	. . . . . 6 ⊢ (∃y(w = ⟨x, y⟩ ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ) ↔ ∃y(w = ⟨x, y⟩ ∧ φ))
7		nfcv 2175	. . . . . . . 8 ⊢ Ⅎy(1st ‘w)
8		nfsbc1v 2776	. . . . . . . 8 ⊢ Ⅎy[(2nd ‘w) / y]φ
9	7, 8	nfsbc 2778	. . . . . . 7 ⊢ Ⅎy[(1st ‘w) / x][(2nd ‘w) / y]φ
10	9	19.41 1573	. . . . . 6 ⊢ (∃y(w = ⟨x, y⟩ ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ) ↔ (∃y w = ⟨x, y⟩ ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ))
11	6, 10	bitr3i 175	. . . . 5 ⊢ (∃y(w = ⟨x, y⟩ ∧ φ) ↔ (∃y w = ⟨x, y⟩ ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ))
12	11	exbii 1493	. . . 4 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃x(∃y w = ⟨x, y⟩ ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ))
13		elvv 4345	. . . . 5 ⊢ (w ∈ (V × V) ↔ ∃x∃y w = ⟨x, y⟩)
14	13	anbi1i 431	. . . 4 ⊢ ((w ∈ (V × V) ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ) ↔ (∃x∃y w = ⟨x, y⟩ ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ))
15	3, 12, 14	3bitr4i 201	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ (w ∈ (V × V) ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ))
16	15	opabbii 3815	. 2 ⊢ {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {⟨w, z⟩ ∣ (w ∈ (V × V) ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ)}
17	1, 16	eqtri 2057	1 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ (w ∈ (V × V) ∧ [(1st ‘w) / x][(2nd ‘w) / y]φ)}

Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  [wsbc 2758  ⟨cop 3370  {copab 3808   × cxp 4286  ‘cfv 4845  {coprab 5456  1^st c1st 5707  2^nd 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-bndl 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