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Definition df-oprab 5528
 Description: Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally x, y, and z are distinct, although the definition doesn't strictly require it. See df-ov 5526 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of the most common operation class builder is given by ov2 in set.mm. (Contributed by SF, 5-Jan-2015.)
Assertion
Ref Expression
df-oprab {x, y, z φ} = {w xyz(w = x, y, z φ)}
Distinct variable groups:   x,w   y,w   z,w   φ,w
Allowed substitution hints:   φ(x,y,z)

Detailed syntax breakdown of Definition df-oprab
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 setvar x
3 vy . . 3 setvar y
4 vz . . 3 setvar z
51, 2, 3, 4coprab 5527 . 2 class {x, y, z φ}
6 vw . . . . . . . . 9 setvar w
76cv 1641 . . . . . . . 8 class w
82cv 1641 . . . . . . . . . 10 class x
93cv 1641 . . . . . . . . . 10 class y
108, 9cop 4561 . . . . . . . . 9 class x, y
114cv 1641 . . . . . . . . 9 class z
1210, 11cop 4561 . . . . . . . 8 class x, y, z
137, 12wceq 1642 . . . . . . 7 wff w = x, y, z
1413, 1wa 358 . . . . . 6 wff (w = x, y, z φ)
1514, 4wex 1541 . . . . 5 wff z(w = x, y, z φ)
1615, 3wex 1541 . . . 4 wff yz(w = x, y, z φ)
1716, 2wex 1541 . . 3 wff xyz(w = x, y, z φ)
1817, 6cab 2339 . 2 class {w xyz(w = x, y, z φ)}
195, 18wceq 1642 1 wff {x, y, z φ} = {w xyz(w = x, y, z φ)}
 Colors of variables: wff setvar class This definition is referenced by:  nfoprab1  5546  nfoprab2  5547  nfoprab3  5548  nfoprab  5549  oprabid  5550  dfoprab2  5558  hboprab1  5559  hboprab2  5560  hboprab3  5561  hboprab  5562  oprabbid  5563  cbvoprab2  5568  eloprabga  5578
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