Theorem dfoprab2 7468

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 2197	. . . 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 2201	. . . . 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 4838	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
4	3	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑣 = ⟨𝑤, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
5	4	pm5.32ri 585	. . . . . . . . . 10 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
6	5	anbi1i 635	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑))
7		anass 473	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
8		an32 658	. . . . . . . . 9 ⊢ (((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
9	6, 7, 8	3bitr3i 304	. . . . . . . 8 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
10	9	exbii 1878	. . . . . . 7 ⊢ (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
11		opex 5445	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
12	11	isseti 3473	. . . . . . . 8 ⊢ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩
13		19.42v 1983	. . . . . . . 8 ⊢ (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩))
14	12, 13	mpbiran2 722	. . . . . . 7 ⊢ (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
15	10, 14	bitri 278	. . . . . 6 ⊢ (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
16	15	3exbii 1880	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
17	2, 16	bitri 278	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
18		19.42vv 1987	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
19	18	2exbii 1879	. . . 4 ⊢ (∃𝑤∃𝑧∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
20	1, 17, 19	3bitr3i 304	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
21	20	abbii 2830	. 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
22		df-oprab 7414	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
23		df-opab 5174	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
24	21, 22, 23	3eqtr4i 2796	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809  {cab 2741  ⟨cop 4595  {copab 5173  {coprab 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-oprab 7414

This theorem is referenced by:  reloprab  7469  oprabv  7470  cbvoprab1  7497  cbvoprab12  7499  cbvoprab3  7501  dmoprab  7513  rnoprab  7515  ssoprab2i  7521  mpomptx  7523  resoprab  7528  funoprabg  7531  elrnmpores  7548  ov6g  7574  dfoprab3s  8046  xpcomco  9051  omxpenlem  9062  nvss  30945  mpomptxf  33023  bj-dfmpoa  37760  mpomptx2  49115