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Axiom ax-rep 4563
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 5671). 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 4568, where you can find some further remarks. A slightly more compact version is shown as axrep2 4565. A quite different variant is zfrep6 6768, which if used in place of ax-rep 4563 would also require that the Separation Scheme axsep 4572 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 8330 and the Boundedness Axiom bnd 8331.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4572, Null Set axnul 4580, and Pairing axpr 4690, 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 4573, ax-nul 4581, and ax-pr 4691 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 1393 . . . . . 6
4 vz . . . . . . 7
54, 2weq 1733 . . . . . 6
63, 5wi 4 . . . . 5
76, 4wal 1393 . . . 4
87, 2wex 1612 . . 3
9 vw . . 3
108, 9wal 1393 . 2
114, 2wel 1819 . . . . 5
12 vx . . . . . . . 8
139, 12wel 1819 . . . . . . 7
1413, 3wa 369 . . . . . 6
1514, 9wex 1612 . . . . 5
1611, 15wb 184 . . . 4
1716, 4wal 1393 . . 3
1817, 2wex 1612 . 2
1910, 18wi 4 1
Colors of variables: wff setvar class
This axiom is referenced by:  axrep1  4564  axnulALT  4579  bj-axrep1  34374  bj-snsetex  34521
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