Axiom ax-rep 1075

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 2716). 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-17 925. Another common variant is derived as zfrep3 1476, where you can find some further remarks. A slightly more compact version is shown as axrep 1473. A quite different variant is zfrep6 2744, which if used in place of ax-rep 1075 would also require that the Separation Scheme zfaus 1480 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 3547 and the Boundedness Axiom bnd 3548. The Collection Principle is sometimes used in place of Replacement as one of the ZF axioms, for example at MathWorld http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html where it is (somewhat misleadingly) called "Replacement".

Assertion

Ref	Expression
ax-rep	⊢ (∀w∃y∀z(∀yφ → z = y) → ∃y∀z(z ∈ y ↔ ∃w(w ∈ x ∧ ∀yφ)))

Distinct variable group(s):   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 672	. . . . . 6 wff ∀yφ
4		vz	. . . . . . 7 set z
5	4, 2	weq 797	. . . . . 6 wff z = y
6	3, 5	wi 2	. . . . 5 wff (∀yφ → z = y)
7	6, 4	wal 672	. . . 4 wff ∀z(∀yφ → z = y)
8	7, 2	wex 678	. . 3 wff ∃y∀z(∀yφ → z = y)
9		vw	. . 3 set w
10	8, 9	wal 672	. 2 wff ∀w∃y∀z(∀yφ → z = y)
11	4, 2	wel 803	. . . . 5 wff z ∈ y
12		vx	. . . . . . . 8 set x
13	9, 12	wel 803	. . . . . . 7 wff w ∈ x
14	13, 3	wa 196	. . . . . 6 wff (w ∈ x ∧ ∀yφ)
15	14, 9	wex 678	. . . . 5 wff ∃w(w ∈ x ∧ ∀yφ)
16	11, 15	wb 127	. . . 4 wff (z ∈ y ↔ ∃w(w ∈ x ∧ ∀yφ))
17	16, 4	wal 672	. . 3 wff ∀z(z ∈ y ↔ ∃w(w ∈ x ∧ ∀yφ))
18	17, 2	wex 678	. 2 wff ∃y∀z(z ∈ y ↔ ∃w(w ∈ x ∧ ∀yφ))
19	10, 18	wi 2	1 wff (∀w∃y∀z(∀yφ → z = y) → ∃y∀z(z ∈ y ↔ ∃w(w ∈ x ∧ ∀yφ)))

This axiom is referenced by:  axrep 1473

metamath.org