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Axiom ax-rep 4597
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 5775). 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 4602, where you can find some further remarks. A slightly more compact version is shown as axrep2 4599. A quite different variant is zfrep6 6901, which if used in place of ax-rep 4597 would also require that the Separation Scheme axsep 4606 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 8512 and the Boundedness Axiom bnd 8513.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 4606, Null Set axnul 4614, and Pairing axpr 4731, 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 4607, ax-nul 4616, and ax-pr 4732 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 1472 . . . . . 6 wff 𝑦𝜑
4 vz . . . . . . 7 setvar 𝑧
54, 2weq 1824 . . . . . 6 wff 𝑧 = 𝑦
63, 5wi 4 . . . . 5 wff (∀𝑦𝜑𝑧 = 𝑦)
76, 4wal 1472 . . . 4 wff 𝑧(∀𝑦𝜑𝑧 = 𝑦)
87, 2wex 1694 . . 3 wff 𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦)
9 vw . . 3 setvar 𝑤
108, 9wal 1472 . 2 wff 𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦)
114, 2wel 1939 . . . . 5 wff 𝑧𝑦
12 vx . . . . . . . 8 setvar 𝑥
139, 12wel 1939 . . . . . . 7 wff 𝑤𝑥
1413, 3wa 382 . . . . . 6 wff (𝑤𝑥 ∧ ∀𝑦𝜑)
1514, 9wex 1694 . . . . 5 wff 𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)
1611, 15wb 194 . . . 4 wff (𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1716, 4wal 1472 . . 3 wff 𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1817, 2wex 1694 . 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  4598  axnulALT  4613  bj-axrep1  31818  bj-snsetex  31976
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