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Axiom ax-rep 4134
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 5297). 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 1605. Another common variant is derived as axrep5 4139, where you can find some further remarks. A slightly more compact version is shown as axrep2 4136. A quite different variant is zfrep6 5711, which if used in place of ax-rep 4134 would also require that the Separation Scheme axsep 4143 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 7558 and the Boundedness Axiom bnd 7559.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4143, Null Set axnul 4151, and Pairing axpr 4214, 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 4144, ax-nul 4152, and ax-pr 4215 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

Assertion
Ref Expression
ax-rep  |-  ( A. w E. y A. z
( A. y ph  ->  z  =  y )  ->  E. y A. z
( z  e.  y  <->  E. w ( w  e.  x  /\  A. y ph ) ) )
Distinct variable group:    x, y, z, w
Allowed substitution groups:    ph( x, y, z, w)

Detailed syntax breakdown of Axiom ax-rep
StepHypRef Expression
1 wph . . . . . . 7  wff  ph
2 vy . . . . . . 7  set  y
31, 2wal 1529 . . . . . 6  wff  A. y ph
4 vz . . . . . . 7  set  z
54, 2weq 1626 . . . . . 6  wff  z  =  y
63, 5wi 6 . . . . 5  wff  ( A. y ph  ->  z  =  y )
76, 4wal 1529 . . . 4  wff  A. z
( A. y ph  ->  z  =  y )
87, 2wex 1530 . . 3  wff  E. y A. z ( A. y ph  ->  z  =  y )
9 vw . . 3  set  w
108, 9wal 1529 . 2  wff  A. w E. y A. z ( A. y ph  ->  z  =  y )
114, 2wel 1688 . . . . 5  wff  z  e.  y
12 vx . . . . . . . 8  set  x
139, 12wel 1688 . . . . . . 7  wff  w  e.  x
1413, 3wa 360 . . . . . 6  wff  ( w  e.  x  /\  A. y ph )
1514, 9wex 1530 . . . . 5  wff  E. w
( w  e.  x  /\  A. y ph )
1611, 15wb 178 . . . 4  wff  ( z  e.  y  <->  E. w
( w  e.  x  /\  A. y ph )
)
1716, 4wal 1529 . . 3  wff  A. z
( z  e.  y  <->  E. w ( w  e.  x  /\  A. y ph ) )
1817, 2wex 1530 . 2  wff  E. y A. z ( z  e.  y  <->  E. w ( w  e.  x  /\  A. y ph ) )
1910, 18wi 6 1  wff  ( A. w E. y A. z
( A. y ph  ->  z  =  y )  ->  E. y A. z
( z  e.  y  <->  E. w ( w  e.  x  /\  A. y ph ) ) )
Colors of variables: wff set class
This axiom is referenced by:  axrep1  4135  axnulALT  4150
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