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Axiom ax-rep 4699
 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 5890). 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 4704, where you can find some further remarks. A slightly more compact version is shown as axrep2 4701. A quite different variant is zfrep6 7027, which if used in place of ax-rep 4699 would also require that the Separation Scheme axsep 4708 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 8637 and the Boundedness Axiom bnd 8638. Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 4708, Null Set axnul 4716, and Pairing axpr 4832, 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 4709, ax-nul 4717, and ax-pr 4833 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 1473 . . . . . 6 wff 𝑦𝜑
4 vz . . . . . . 7 setvar 𝑧
54, 2weq 1861 . . . . . 6 wff 𝑧 = 𝑦
63, 5wi 4 . . . . 5 wff (∀𝑦𝜑𝑧 = 𝑦)
76, 4wal 1473 . . . 4 wff 𝑧(∀𝑦𝜑𝑧 = 𝑦)
87, 2wex 1695 . . 3 wff 𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦)
9 vw . . 3 setvar 𝑤
108, 9wal 1473 . 2 wff 𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦)
114, 2wel 1978 . . . . 5 wff 𝑧𝑦
12 vx . . . . . . . 8 setvar 𝑥
139, 12wel 1978 . . . . . . 7 wff 𝑤𝑥
1413, 3wa 383 . . . . . 6 wff (𝑤𝑥 ∧ ∀𝑦𝜑)
1514, 9wex 1695 . . . . 5 wff 𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)
1611, 15wb 195 . . . 4 wff (𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1716, 4wal 1473 . . 3 wff 𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
1817, 2wex 1695 . 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  4700  axnulALT  4715  bj-axrep1  31976  bj-snsetex  32144
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