Axiom ax-rep 5213

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 6584). 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 5221, where you can find some further remarks. A slightly more compact version is shown as axrep2 5216. A quite different variant is zfrep6 5225, which if used in place of ax-rep 5213 would also require that the Separation Scheme axsep 5231 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 9812 and the Boundedness Axiom bnd 9813.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5231, Null Set axnul 5241, and Pairing axpr 5368, 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 5232, ax-nul 5242, and ax-pr 5374 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

Assertion

Ref	Expression
ax-rep	⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Detailed syntax breakdown of Axiom ax-rep

Step	Hyp	Ref	Expression
1		wph	. . . . . . 7 wff 𝜑
2		vy	. . . . . . 7 setvar 𝑦
3	1, 2	wal 1540	. . . . . 6 wff ∀𝑦𝜑
4		vz	. . . . . . 7 setvar 𝑧
5	4, 2	weq 1964	. . . . . 6 wff 𝑧 = 𝑦
6	3, 5	wi 4	. . . . 5 wff (∀𝑦𝜑 → 𝑧 = 𝑦)
7	6, 4	wal 1540	. . . 4 wff ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
8	7, 2	wex 1781	. . 3 wff ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
9		vw	. . 3 setvar 𝑤
10	8, 9	wal 1540	. 2 wff ∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
11	4, 2	wel 2115	. . . . 5 wff 𝑧 ∈ 𝑦
12		vx	. . . . . . . 8 setvar 𝑥
13	9, 12	wel 2115	. . . . . . 7 wff 𝑤 ∈ 𝑥
14	13, 3	wa 395	. . . . . 6 wff (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
15	14, 9	wex 1781	. . . . 5 wff ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
16	11, 15	wb 206	. . . 4 wff (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
17	16, 4	wal 1540	. . 3 wff ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
18	17, 2	wex 1781	. 2 wff ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
19	10, 18	wi 4	1 wff (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))

This axiom is referenced by:  axrep1  5214  axrep4v  5218  axrep4  5219  axrep6OLD  5223  axnulALT  5240  axnulALT2  35221  bj-snsetex  37267