Theorem dfopab2 7865

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 3731	. . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
2	1	19.41 2231	. . . 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 7863	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑 ↔ 𝜑))
4	3	pm5.32i 574	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	exbii 1851	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑧)
7		nfsbc1v 3731	. . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑧) / 𝑦]𝜑
8	6, 7	nfsbcw 3733	. . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
9	8	19.41 2231	. . . . . 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 276	. . . . 5 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
11	10	exbii 1851	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
12		elvv 5652	. . . . 5 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
13	12	anbi1i 623	. . . 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 302	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
15	14	abbii 2809	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑)}
16		df-opab 5133	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		df-rab 3072	. 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 2776	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑}

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  {crab 3067  Vcvv 3422  [wsbc 3711  ⟨cop 4564  {copab 5132   × cxp 5578  ‘cfv 6418  1^st c1st 7802  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-1st 7804  df-2nd 7805

This theorem is referenced by: (None)