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Axiom ax-rep 4378
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 5466). Although may be any wff whatsoever, this axiom is useful (i.e. its antecedent is satisfied) when we are given some function and encodes the predicate "the value of the function at is ." Thus, will ordinarily have free variables and - think of it informally as ( , ). We prefix with the quantifier A. in order to "protect" the axiom from any containing , thus allowing us to eliminate any restrictions on . Another common variant is derived as axrep5 4383, where you can find some further remarks. A slightly more compact version is shown as axrep2 4380. A quite different variant is zfrep6 6514, which if used in place of ax-rep 4378 would also require that the Separation Scheme axsep 4387 be stated as a separate axiom.

There is very a strong generalization of Replacement that doesn't demand function-like behavior of . Two versions of this generalization are called the Collection Principle cp 8045 and the Boundedness Axiom bnd 8046.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4387, Null Set axnul 4395, and Pairing axpr 4502, 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 4388, ax-nul 4396, and ax-pr 4503 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

Assertion
Ref Expression
ax-rep
Distinct variable group:   , , ,

Detailed syntax breakdown of Axiom ax-rep
StepHypRef Expression
1 wph . . . . . . 7
2 vy . . . . . . 7
31, 2wal 1580 . . . . . 6
4 vz . . . . . . 7
54, 2weq 1688 . . . . . 6
63, 5wi 4 . . . . 5
76, 4wal 1580 . . . 4
87, 2wex 1581 . . 3
9 vw . . 3
108, 9wal 1580 . 2
114, 2wel 1750 . . . . 5
12 vx . . . . . . . 8
139, 12wel 1750 . . . . . . 7
1413, 3wa 362 . . . . . 6
1514, 9wex 1581 . . . . 5
1611, 15wb 178 . . . 4
1716, 4wal 1580 . . 3
1817, 2wex 1581 . 2
1910, 18wi 4 1
Colors of variables: wff setvar class
This axiom is referenced by:  axrep1  4379  axnulALT  4394  bj-axrep1  31803  bj-snsetex  31903
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