Theorem dfoprab2 6078

Description: Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)

Assertion

Ref	Expression
dfoprab2	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem dfoprab2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		excom 1712	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤∃𝑧∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2		exrot4 1739	. . . . 5 ⊢ (∃𝑧∃𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3		opeq1 3867	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
4	3	eqeq2d 2243	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑣 = ⟨𝑤, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
5	4	pm5.32ri 455	. . . . . . . . . 10 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
6	5	anbi1i 458	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑))
7		anass 401	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
8		an32 564	. . . . . . . . 9 ⊢ (((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
9	6, 7, 8	3bitr3i 210	. . . . . . . 8 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
10	9	exbii 1654	. . . . . . 7 ⊢ (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
11		vex 2806	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		vex 2806	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
13	11, 12	opex 4327	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
14	13	isseti 2812	. . . . . . . 8 ⊢ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩
15		19.42v 1955	. . . . . . . 8 ⊢ (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩))
16	14, 15	mpbiran2 950	. . . . . . 7 ⊢ (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
17	10, 16	bitri 184	. . . . . 6 ⊢ (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
18	17	3exbii 1656	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
19	2, 18	bitri 184	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
20		19.42vv 1960	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
21	20	2exbii 1655	. . . 4 ⊢ (∃𝑤∃𝑧∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
22	1, 19, 21	3bitr3i 210	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
23	22	abbii 2347	. 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
24		df-oprab 6032	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
25		df-opab 4156	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
26	23, 24, 25	3eqtr4i 2262	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541  {cab 2217  ⟨cop 3676  {copab 4154  {coprab 6029

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-oprab 6032

This theorem is referenced by:  reloprab  6079  cbvoprab1  6103  cbvoprab12  6105  cbvoprab3  6107  dmoprab  6112  rnoprab  6114  ssoprab2i  6120  mpomptx  6122  resoprab  6127  funoprabg  6130  ov6g  6170  dfoprab3s  6362  xpcomco  7053