Axiom ax-rep 2661

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 3516). Although ph may be any wff whatsoever, this axiom is useful (i.e. its antecedent is satisfied) when we are given some function and ph encodes the predicate "the value of the function at w is z". Thus ph will ordinarily have free variables w and z - think of it informally as ph(w, z). We prefix ph with the quantifier A.y in order to "protect" the axiom from any ph containing y, thus allowing us to eliminate any restrictions on ph. This makes the axiom usable in a formalization that omits the logically redundant axiom ax-17 1190. Another common variant is derived as axrep5 2666, where you can find some further remarks. A slightly more compact version is shown as axrep2 2663. A quite different variant is zfrep6 3554, which if used in place of ax-rep 2661 would also require that the Separation Scheme axsep 2670 be stated as a separate axiom.

There is very a strong generalization of Replacement that doesn't demand function-like behavior of ph. Two versions of this generalization are called the Collection Principle cp 4646 and the Boundedness Axiom bnd 4647.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 2670, Null Set axnul 2677, and Pairing axpr 2746, 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 2671, ax-nul 2678, and ax-pr 2747 below the theorems that prove them.

Assertion

Ref	Expression
ax-rep	|- (A.wE.yA.z(A.yph -> z = y) -> E.yA.z(z e. y <-> E.w(w e. x /\ A.yph)))

Distinct variable group:   x,y,z,w

Detailed syntax breakdown of Axiom ax-rep

Step	Hyp	Ref	Expression
1		wph	. . . . . . 7 wff ph
2		vy	. . . . . . 7 set y
3	1, 2	wal 950	. . . . . 6 wff A.yph
4		vz	. . . . . . . 8 set z
5	4	cv 1098	. . . . . . 7 class z
6	2	cv 1098	. . . . . . 7 class y
7	5, 6	wceq 1099	. . . . . 6 wff z = y
8	3, 7	wi 3	. . . . 5 wff (A.yph -> z = y)
9	8, 4	wal 950	. . . 4 wff A.z(A.yph -> z = y)
10	9, 2	wex 956	. . 3 wff E.yA.z(A.yph -> z = y)
11		vw	. . 3 set w
12	10, 11	wal 950	. 2 wff A.wE.yA.z(A.yph -> z = y)
13	5, 6	wcel 1105	. . . . 5 wff z e. y
14	11	cv 1098	. . . . . . . 8 class w
15		vx	. . . . . . . . 9 set x
16	15	cv 1098	. . . . . . . 8 class x
17	14, 16	wcel 1105	. . . . . . 7 wff w e. x
18	17, 3	wa 223	. . . . . 6 wff (w e. x /\ A.yph)
19	18, 11	wex 956	. . . . 5 wff E.w(w e. x /\ A.yph)
20	13, 19	wb 146	. . . 4 wff (z e. y <-> E.w(w e. x /\ A.yph))
21	20, 4	wal 950	. . 3 wff A.z(z e. y <-> E.w(w e. x /\ A.yph))
22	21, 2	wex 956	. 2 wff E.yA.z(z e. y <-> E.w(w e. x /\ A.yph))
23	12, 22	wi 3	1 wff (A.wE.yA.z(A.yph -> z = y) -> E.yA.z(z e. y <-> E.w(w e. x /\ A.yph)))

This axiom is referenced by:  axrep1 2662  axnul2 2676

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