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Axiom ax-rep 4731
Description: Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory. Axiom 5 of [TakeutiZaring] p. 19. It tells us that the image of any set under a function is also a set (see the variant funimaex 5934). Although 𝜑 may be any wff whatsoever, this axiom is useful (i.e. its antecedent is satisfied) when we are given some function and 𝜑 encodes the predicate "the value of the function at 𝑤 is 𝑧." Thus, 𝜑 will ordinarily have free variables 𝑤 and 𝑧- think of it informally as 𝜑(𝑤, 𝑧). We prefix 𝜑 with the quantifier 𝑦 in order to "protect" the axiom from any 𝜑 containing 𝑦, thus allowing us to eliminate any restrictions on 𝜑. Another common variant is derived as axrep5 4736, where you can find some further remarks. A slightly more compact version is shown as axrep2 4733. A quite different variant is zfrep6 7081, which if used in place of ax-rep 4731 would also require that the Separation Scheme axsep 4740 be stated as a separate axiom.

There is a very strong generalization of Replacement that doesn't demand function-like behavior of 𝜑. Two versions of this generalization are called the Collection Principle cp 8698 and the Boundedness Axiom bnd 8699.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 4740, Null Set axnul 4748, and Pairing axpr 4866, all of which we derive from Replacement. In order to make it easier to identify the uses of those redundant axioms, we restate them as axioms ax-sep 4741, ax-nul 4749, and ax-pr 4867 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

Assertion
Ref Expression
ax-rep (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Detailed syntax breakdown of Axiom ax-rep
StepHypRef Expression
1 wph . . . . . . 7 wff 𝜑
2 vy . . . . . . 7 setvar 𝑦
31, 2wal 1478 . . . . . 6 wff 𝑦𝜑
4 vz . . . . . . 7 setvar 𝑧
54, 2weq 1871 . . . . . 6 wff 𝑧 = 𝑦
63, 5wi 4 . . . . 5 wff (∀𝑦𝜑𝑧 = 𝑦)
76, 4wal 1478 . . . 4 wff 𝑧(∀𝑦𝜑𝑧 = 𝑦)
87, 2wex 1701 . . 3 wff 𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦)
9 vw . . 3 setvar 𝑤
108, 9wal 1478 . 2 wff 𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦)
114, 2wel 1988 . . . . 5 wff 𝑧𝑦
12 vx . . . . . . . 8 setvar 𝑥
139, 12wel 1988 . . . . . . 7 wff 𝑤𝑥
1413, 3wa 384 . . . . . 6 wff (𝑤𝑥 ∧ ∀𝑦𝜑)
1514, 9wex 1701 . . . . 5 wff 𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)
1611, 15wb 196 . . . 4 wff (𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1716, 4wal 1478 . . 3 wff 𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1817, 2wex 1701 . 2 wff 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1910, 18wi 4 1 wff (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
This axiom is referenced by:  axrep1  4732  axnulALT  4747  bj-axrep1  32428  bj-snsetex  32595
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