Theorem dfoprab3s 6081

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 5811	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfsbc1v 2922	. . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑
3	2	19.41 1664	. . . 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 6078	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 ↔ 𝜑))
5	4	pm5.32i 449	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	5	exbii 1584	. . . . . 6 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7		nfcv 2279	. . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑤)
8		nfsbc1v 2922	. . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑤) / 𝑦]𝜑
9	7, 8	nfsbc 2924	. . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑
10	9	19.41 1664	. . . . . 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 185	. . . . 5 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
12	11	exbii 1584	. . . 4 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
13		elvv 4596	. . . . 5 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
14	13	anbi1i 453	. . . 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 211	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
16	15	opabbii 3990	. 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 2158	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)}

Syntax hints:   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2681  [wsbc 2904  ⟨cop 3525  {copab 3983   × cxp 4532  ‘cfv 5118  {coprab 5768  1^st c1st 6029  2^nd c2nd 6030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-oprab 5771  df-1st 6031  df-2nd 6032

This theorem is referenced by:  dfoprab3  6082