Axiom ax-rep 2708

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 3592). 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-17 975. Another common variant is derived as axrep5 2713, where you can find some further remarks. A slightly more compact version is shown as axrep2 2710. A quite different variant is zfrep6 3630, which if used in place of ax-rep 2708 would also require that the Separation Scheme axsep 2717 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 4739 and the Boundedness Axiom bnd 4740.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 2717, Null Set axnul 2724, and Pairing axpr 2794, 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 2718, ax-nul 2725, and ax-pr 2795 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 958	. . . . . 6 wff ∀yφ
4		vz	. . . . . . . 8 set z
5	4	cv 959	. . . . . . 7 class z
6	2	cv 959	. . . . . . 7 class y
7	5, 6	wceq 960	. . . . . 6 wff z = y
8	3, 7	wi 3	. . . . 5 wff (∀yφ → z = y)
9	8, 4	wal 958	. . . 4 wff ∀z(∀yφ → z = y)
10	9, 2	wex 984	. . 3 wff ∃y∀z(∀yφ → z = y)
11		vw	. . 3 set w
12	10, 11	wal 958	. 2 wff ∀w∃y∀z(∀yφ → z = y)
13	5, 6	wcel 962	. . . . 5 wff z ∈ y
14	11	cv 959	. . . . . . . 8 class w
15		vx	. . . . . . . . 9 set x
16	15	cv 959	. . . . . . . 8 class x
17	14, 16	wcel 962	. . . . . . 7 wff w ∈ x
18	17, 3	wa 223	. . . . . 6 wff (w ∈ x ⋀ ∀yφ)
19	18, 11	wex 984	. . . . 5 wff ∃w(w ∈ x ⋀ ∀yφ)
20	13, 19	wb 146	. . . 4 wff (z ∈ y ↔ ∃w(w ∈ x ⋀ ∀yφ))
21	20, 4	wal 958	. . 3 wff ∀z(z ∈ y ↔ ∃w(w ∈ x ⋀ ∀yφ))
22	21, 2	wex 984	. 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 2709  axnul2 2723

Copyright terms: Public domain