Definition df-opab 5137

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 5517 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 7994. An example is given by ex-opab 30490. (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 5136	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5		vz	. . . . . . . 8 setvar 𝑧
6	5	cv 1541	. . . . . . 7 class 𝑧
7	2	cv 1541	. . . . . . . 8 class 𝑥
8	3	cv 1541	. . . . . . . 8 class 𝑦
9	7, 8	cop 4563	. . . . . . 7 class ⟨𝑥, 𝑦⟩
10	6, 9	wceq 1542	. . . . . 6 wff 𝑧 = ⟨𝑥, 𝑦⟩
11	10, 1	wa 395	. . . . 5 wff (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
12	11, 3	wex 1781	. . . 4 wff ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
13	12, 2	wex 1781	. . 3 wff ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
14	13, 5	cab 2713	. 2 class {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15	4, 14	wceq 1542	1 wff {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

This definition is referenced by:  opabss  5138  opabbid  5139  opabbidv  5140  nfopabd  5142  nfopab1  5144  nfopab2  5145  cbvopab  5146  cbvopabv  5147  cbvopab1  5148  cbvopab1g  5149  cbvopab2  5150  cbvopab1s  5151  cbvopab1v  5152  cbvopab2v  5153  unopab  5154  opabidw  5468  opabid  5469  elopabw  5470  ssopab2  5490  iunopab  5503  dfid2  5517  dfid3  5518  elxpi  5642  opabssxpd  5667  rabxp  5668  csbxp  5721  relopabi  5767  relopabiALT  5768  cnv0OLD  6093  dfoprab2  7414  dmoprab  7459  dfopab2  7994  brdom7disj  10442  brdom6disj  10443  opabssi  32674  cbvopab1davw  36434  cbvopab2davw  36435  cbvopabdavw  36436  bj-dfid2ALT  37360  rnxrn  38730  dropab1  44861  dropab2  44862  csbxpgVD  45308  relopabVD  45315