Theorem dfoprab3s 5836

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 5572	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfsbc1v 2833	. . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑
3	2	19.41 1616	. . . 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 5833	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 ↔ 𝜑))
5	4	pm5.32i 441	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	5	exbii 1536	. . . . . 6 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7		nfcv 2219	. . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑤)
8		nfsbc1v 2833	. . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑤) / 𝑦]𝜑
9	7, 8	nfsbc 2835	. . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑
10	9	19.41 1616	. . . . . 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 184	. . . . 5 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
12	11	exbii 1536	. . . 4 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
13		elvv 4420	. . . . 5 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
14	13	anbi1i 445	. . . 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 210	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
16	15	opabbii 3845	. 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 2101	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)}

Syntax hints:   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433  Vcvv 2601  [wsbc 2815  ⟨cop 3401  {copab 3838   × cxp 4361  ‘cfv 4922  {coprab 5533  1^st c1st 5785  2^nd c2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fv 4930  df-oprab 5536  df-1st 5787  df-2nd 5788

This theorem is referenced by:  dfoprab3  5837