Definition df-opab 5170

Description: Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually 𝑥 and 𝑦 are distinct, although the definition does not require it (see dfid2 5535 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also called "class builder" in the literature. An alternate definition using no existential quantifiers is shown by dfopab2 8031. An example is given by ex-opab 30361. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
df-opab	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Detailed syntax breakdown of Definition df-opab

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4	1, 2, 3	copab 5169	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5		vz	. . . . . . . 8 setvar 𝑧
6	5	cv 1539	. . . . . . 7 class 𝑧
7	2	cv 1539	. . . . . . . 8 class 𝑥
8	3	cv 1539	. . . . . . . 8 class 𝑦
9	7, 8	cop 4595	. . . . . . 7 class ⟨𝑥, 𝑦⟩
10	6, 9	wceq 1540	. . . . . 6 wff 𝑧 = ⟨𝑥, 𝑦⟩
11	10, 1	wa 395	. . . . 5 wff (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
12	11, 3	wex 1779	. . . 4 wff ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
13	12, 2	wex 1779	. . 3 wff ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
14	13, 5	cab 2707	. 2 class {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15	4, 14	wceq 1540	1 wff {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

This definition is referenced by:  opabss  5171  opabbid  5172  opabbidv  5173  nfopabd  5175  nfopab1  5177  nfopab2  5178  cbvopab  5179  cbvopabv  5180  cbvopab1  5181  cbvopab1g  5182  cbvopab2  5183  cbvopab1s  5184  cbvopab1v  5185  cbvopab2v  5186  unopab  5187  opabidw  5484  opabid  5485  elopabw  5486  ssopab2  5506  iunopab  5519  iunopabOLD  5520  dfid2  5535  dfid3  5536  elxpi  5660  opabssxpd  5685  rabxp  5686  csbxp  5738  ssrelOLD  5746  relopabi  5785  relopabiALT  5786  cnv0  6113  dfoprab2  7447  dmoprab  7492  dfopab2  8031  brdom7disj  10484  brdom6disj  10485  opabssi  32541  cbvopab1davw  36252  cbvopab2davw  36253  cbvopabdavw  36254  bj-dfid2ALT  37053  rnxrn  38384  dropab1  44436  dropab2  44437  csbxpgVD  44883  relopabVD  44890