Theorem dfopab2 6274

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 3016	. . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
2	1	19.41 1708	. . . 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 6272	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑 ↔ 𝜑))
4	3	pm5.32i 454	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	exbii 1627	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6		nfcv 2347	. . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑧)
7		nfsbc1v 3016	. . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑧) / 𝑦]𝜑
8	6, 7	nfsbc 3018	. . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑
9	8	19.41 1708	. . . . . 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 186	. . . . 5 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
11	10	exbii 1627	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
12		elvv 4736	. . . . 5 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
13	12	anbi1i 458	. . . 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 212	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑))
15	14	abbii 2320	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑)}
16		df-opab 4105	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		df-rab 2492	. 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 2235	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑}

Syntax hints:   ∧ wa 104   = wceq 1372  ∃wex 1514   ∈ wcel 2175  {cab 2190  {crab 2487  Vcvv 2771  [wsbc 2997  ⟨cop 3635  {copab 4103   × cxp 4672  ‘cfv 5270  1^st c1st 6223  2^nd c2nd 6224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fv 5278  df-1st 6225  df-2nd 6226

This theorem is referenced by: (None)