Theorem dfoprab2 7416

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 2167	. . . 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 2171	. . . . 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 4829	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
4	3	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑣 = ⟨𝑤, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
5	4	pm5.32ri 575	. . . . . . . . . 10 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
6	5	anbi1i 624	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑))
7		anass 468	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
8		an32 646	. . . . . . . . 9 ⊢ (((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
9	6, 7, 8	3bitr3i 301	. . . . . . . 8 ⊢ ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
10	9	exbii 1849	. . . . . . 7 ⊢ (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
11		opex 5412	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
12	11	isseti 3458	. . . . . . . 8 ⊢ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩
13		19.42v 1954	. . . . . . . 8 ⊢ (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩))
14	12, 13	mpbiran2 710	. . . . . . 7 ⊢ (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
15	10, 14	bitri 275	. . . . . 6 ⊢ (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
16	15	3exbii 1851	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
17	2, 16	bitri 275	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
18		19.42vv 1958	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
19	18	2exbii 1850	. . . 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 301	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
21	20	abbii 2803	. 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 7362	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
23		df-opab 5161	. 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 2769	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780  {cab 2714  ⟨cop 4586  {copab 5160  {coprab 7359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-opab 5161  df-oprab 7362

This theorem is referenced by:  reloprab  7417  oprabv  7418  cbvoprab1  7445  cbvoprab12  7447  cbvoprab3  7449  dmoprab  7461  rnoprab  7463  ssoprab2i  7469  mpomptx  7471  resoprab  7476  funoprabg  7479  elrnmpores  7496  ov6g  7522  dfoprab3s  7997  xpcomco  8995  omxpenlem  9006  nvss  30668  mpomptxf  32757  bj-dfmpoa  37323  mpomptx2  48581