Theorem dfopab2 6157

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) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑}

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dfopab2

Step	Hyp	Ref	Expression
1		nfsbc1v 2969	. . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
2	1	19.41 1674	. . . 4 ⊢ (∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
3		sbcopeq1a 6155	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑 ↔ 𝜑))
4	3	pm5.32i 450	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	exbii 1593	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑧)
7		nfsbc1v 2969	. . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑧) / 𝑦]𝜑
8	6, 7	nfsbc 2971	. . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
9	8	19.41 1674	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
10	5, 9	bitr3i 185	. . . . 5 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
11	10	exbii 1593	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
12		elvv 4666	. . . . 5 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
13	12	anbi1i 454	. . . 4 ⊢ ((𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
14	2, 11, 13	3bitr4i 211	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
15	14	abbii 2282	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑)}
16		df-opab 4044	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		df-rab 2453	. 2 ⊢ {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑)}
18	15, 16, 17	3eqtr4i 2196	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑}

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {cab 2151  {crab 2448  Vcvv 2726  [wsbc 2951  ⟨cop 3579  {copab 4042   × cxp 4602  ‘cfv 5188  1^st c1st 6106  2^nd c2nd 6107

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-1st 6108  df-2nd 6109

This theorem is referenced by: (None)