Axiom ax-rep 5190

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 6441). 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 5196, where you can find some further remarks. A slightly more compact version is shown as axrep2 5193. A quite different variant is zfrep6 7656, which if used in place of ax-rep 5190 would also require that the Separation Scheme axsep 5202 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 9320 and the Boundedness Axiom bnd 9321.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5202, Null Set axnul 5209, and Pairing axpr 5329, 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 5203, ax-nul 5210, and ax-pr 5330 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 1535	. . . . . 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 1535	. . . 4 wff ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
8	7, 2	wex 1780	. . 3 wff ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
9		vw	. . 3 setvar 𝑤
10	8, 9	wal 1535	. 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 398	. . . . . 6 wff (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
15	14, 9	wex 1780	. . . . 5 wff ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
16	11, 15	wb 208	. . . 4 wff (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
17	16, 4	wal 1535	. . 3 wff ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
18	17, 2	wex 1780	. 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  5191  axrep6  5197  axnulALT  5208  bj-snsetex  34278