Axiom ax-rep 5236

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 6624). 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 5244, where you can find some further remarks. A slightly more compact version is shown as axrep2 5239. A quite different variant is zfrep6 5248, which if used in place of ax-rep 5236 would also require that the Separation Scheme axsep 5254 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 9896 and the Boundedness Axiom bnd 9897.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5254, Null Set axnul 5266, and Pairing axpr 5396, 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 5255, ax-nul 5267, and ax-pr 5402 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 1568	. . . . . 6 wff ∀𝑦𝜑
4		vz	. . . . . . 7 setvar 𝑧
5	4, 2	weq 1995	. . . . . 6 wff 𝑧 = 𝑦
6	3, 5	wi 4	. . . . 5 wff (∀𝑦𝜑 → 𝑧 = 𝑦)
7	6, 4	wal 1568	. . . 4 wff ∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
8	7, 2	wex 1812	. . 3 wff ∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
9		vw	. . 3 setvar 𝑤
10	8, 9	wal 1568	. 2 wff ∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
11	4, 2	wel 2146	. . . . 5 wff 𝑧 ∈ 𝑦
12		vx	. . . . . . . 8 setvar 𝑥
13	9, 12	wel 2146	. . . . . . 7 wff 𝑤 ∈ 𝑥
14	13, 3	wa 401	. . . . . 6 wff (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
15	14, 9	wex 1812	. . . . 5 wff ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
16	11, 15	wb 209	. . . 4 wff (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
17	16, 4	wal 1568	. . 3 wff ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
18	17, 2	wex 1812	. 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 used by:  axrep1  5237  axrep4v  5241  axrep4  5242  axrep6OLD  5246  axnulALT  5265  axnulALT2  35575  bj-snsetex  37692