Axiom ax-rep 5237

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 6623). 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 5245, where you can find some further remarks. A slightly more compact version is shown as axrep2 5240. A quite different variant is zfrep6 5249, which if used in place of ax-rep 5237 would also require that the Separation Scheme axsep 5255 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 9876 and the Boundedness Axiom bnd 9877.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5255, Null Set axnul 5267, and Pairing axpr 5398, 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 5256, ax-nul 5268, and ax-pr 5404 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 1566	. . . . . 6 wff ∀𝑦𝜑
4		vz	. . . . . . 7 setvar 𝑧
5	4, 2	weq 1990	. . . . . 6 wff 𝑧 = 𝑦
6	3, 5	wi 4	. . . . 5 wff (∀𝑦𝜑 → 𝑧 = 𝑦)
7	6, 4	wal 1566	. . . 4 wff ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
8	7, 2	wex 1807	. . 3 wff ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
9		vw	. . 3 setvar 𝑤
10	8, 9	wal 1566	. 2 wff ∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
11	4, 2	wel 2142	. . . . 5 wff 𝑧 ∈ 𝑦
12		vx	. . . . . . . 8 setvar 𝑥
13	9, 12	wel 2142	. . . . . . 7 wff 𝑤 ∈ 𝑥
14	13, 3	wa 400	. . . . . 6 wff (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
15	14, 9	wex 1807	. . . . 5 wff ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
16	11, 15	wb 209	. . . 4 wff (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
17	16, 4	wal 1566	. . 3 wff ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
18	17, 2	wex 1807	. 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  5238  axrep4v  5242  axrep4  5243  axrep6OLD  5247  axnulALT  5266  axnulALT2  35436  bj-snsetex  37543