Axiom ax-rep 5284

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 6633). 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 5290, where you can find some further remarks. A slightly more compact version is shown as axrep2 5287. A quite different variant is zfrep6 7936, which if used in place of ax-rep 5284 would also require that the Separation Scheme axsep 5297 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 9882 and the Boundedness Axiom bnd 9883.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5297, Null Set axnul 5304, and Pairing axpr 5425, 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 5298, ax-nul 5305, and ax-pr 5426 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 1967	. . . . . 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 1782	. . 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 2108	. . . . 5 wff 𝑧 ∈ 𝑦
12		vx	. . . . . . . 8 setvar 𝑥
13	9, 12	wel 2108	. . . . . . 7 wff 𝑤 ∈ 𝑥
14	13, 3	wa 397	. . . . . 6 wff (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
15	14, 9	wex 1782	. . . . 5 wff ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
16	11, 15	wb 205	. . . 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 1782	. 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  5285  axrep6  5291  axnulALT  5303  bj-snsetex  35782