Axiom ax-rep 2688

Description: Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory. Axiom 5 of [TakeutiZaring] p. 19. It tells us that that the image of any set under a function is also a set (see the variant funimaex 3568). 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. This makes the axiom usable in a formalization that omits the logically redundant axiom ax-17 969. Another common variant is derived as axrep5 2693, where you can find some further remarks. A slightly more compact version is shown as axrep2 2690. A quite different variant is zfrep6 3606, which if used in place of ax-rep 2688 would also require that the Separation Scheme axsep 2697 be stated as a separate axiom.

There is very a strong generalization of Replacement that doesn't demand function-like behavior of φ. Two versions of this generalization are called the Collection Principle cp 4702 and the Boundedness Axiom bnd 4703.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 2697, Null Set axnul 2704, and Pairing axpr 2773, 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 2698, ax-nul 2705, and ax-pr 2774 below the theorems that prove them.

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

Detailed syntax breakdown of Axiom ax-rep

Step	Hyp	Ref	Expression
1		wph	. . . . . . 7 wff φ
2		vy	. . . . . . 7 set y
3	1, 2	wal 952	. . . . . 6 wff ∀yφ
4		vz	. . . . . . . 8 set z
5	4	cv 953	. . . . . . 7 class z
6	2	cv 953	. . . . . . 7 class y
7	5, 6	wceq 954	. . . . . 6 wff z = y
8	3, 7	wi 3	. . . . 5 wff (∀yφ → z = y)
9	8, 4	wal 952	. . . 4 wff ∀z(∀yφ → z = y)
10	9, 2	wex 978	. . 3 wff ∃y∀z(∀yφ → z = y)
11		vw	. . 3 set w
12	10, 11	wal 952	. 2 wff ∀w∃y∀z(∀yφ → z = y)
13	5, 6	wcel 956	. . . . 5 wff z ∈ y
14	11	cv 953	. . . . . . . 8 class w
15		vx	. . . . . . . . 9 set x
16	15	cv 953	. . . . . . . 8 class x
17	14, 16	wcel 956	. . . . . . 7 wff w ∈ x
18	17, 3	wa 223	. . . . . 6 wff (w ∈ x ⋀ ∀yφ)
19	18, 11	wex 978	. . . . 5 wff ∃w(w ∈ x ⋀ ∀yφ)
20	13, 19	wb 146	. . . 4 wff (z ∈ y ↔ ∃w(w ∈ x ⋀ ∀yφ))
21	20, 4	wal 952	. . 3 wff ∀z(z ∈ y ↔ ∃w(w ∈ x ⋀ ∀yφ))
22	21, 2	wex 978	. 2 wff ∃y∀z(z ∈ y ↔ ∃w(w ∈ x ⋀ ∀yφ))
23	12, 22	wi 3	1 wff (∀w∃y∀z(∀yφ → z = y) → ∃y∀z(z ∈ y ↔ ∃w(w ∈ x ⋀ ∀yφ)))

This axiom is referenced by:  axrep1 2689  axnul2 2703

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