Theorem dfopab2 8049

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 3773	. . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
2	1	19.41 2277	. . . 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 8046	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑 ↔ 𝜑))
4	3	pm5.32i 584	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	exbii 1875	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑧)
7		nfsbc1v 3773	. . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑧) / 𝑦]𝜑
8	6, 7	nfsbcw 3775	. . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
9	8	19.41 2277	. . . . . 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 280	. . . . 5 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
11	10	exbii 1875	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
12		elvv 5737	. . . . 5 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
13	12	anbi1i 635	. . . 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 306	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
15	14	abbii 2836	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑)}
16		df-opab 5178	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		df-rab 3424	. 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 2802	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑}

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  {crab 3423  Vcvv 3463  [wsbc 3753  ⟨cop 4600  {copab 5177   × cxp 5660  ‘cfv 6537  1^st c1st 7984  2^nd c2nd 7985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7986  df-2nd 7987

This theorem is referenced by: (None)