Theorem dfopab2 8063

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 3759	. . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
2	1	19.41 2272	. . . 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 8060	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑 ↔ 𝜑))
4	3	pm5.32i 585	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	exbii 1881	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑧)
7		nfsbc1v 3759	. . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑧) / 𝑦]𝜑
8	6, 7	nfsbcw 3761	. . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
9	8	19.41 2272	. . . . . 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 1881	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
12		elvv 5726	. . . . 5 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
13	12	anbi1i 636	. . . 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 2828	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑)}
16		df-opab 5168	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		df-rab 3414	. 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 2794	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  {crab 3413  Vcvv 3451  [wsbc 3739  ⟨cop 4590  {copab 5167   × cxp 5649  ‘cfv 6538  1^st c1st 7999  2^nd c2nd 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 8001  df-2nd 8002

This theorem is used by: (None)