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Axiom ax-rep 4312
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 5523). 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 1626. Another common variant is derived as axrep5 4317, where you can find some further remarks. A slightly more compact version is shown as axrep2 4314. A quite different variant is zfrep6 5960, which if used in place of ax-rep 4312 would also require that the Separation Scheme axsep 4321 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 7807 and the Boundedness Axiom bnd 7808.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4321, Null Set axnul 4329, and Pairing axpr 4394, 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 4322, ax-nul 4330, and ax-pr 4395 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 hints:    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 1549 . . . . . 6  wff  A. y ph
4 vz . . . . . . 7  set  z
54, 2weq 1653 . . . . . 6  wff  z  =  y
63, 5wi 4 . . . . 5  wff  ( A. y ph  ->  z  =  y )
76, 4wal 1549 . . . 4  wff  A. z
( A. y ph  ->  z  =  y )
87, 2wex 1550 . . 3  wff  E. y A. z ( A. y ph  ->  z  =  y )
9 vw . . 3  set  w
108, 9wal 1549 . 2  wff  A. w E. y A. z ( A. y ph  ->  z  =  y )
114, 2wel 1726 . . . . 5  wff  z  e.  y
12 vx . . . . . . . 8  set  x
139, 12wel 1726 . . . . . . 7  wff  w  e.  x
1413, 3wa 359 . . . . . 6  wff  ( w  e.  x  /\  A. y ph )
1514, 9wex 1550 . . . . 5  wff  E. w
( w  e.  x  /\  A. y ph )
1611, 15wb 177 . . . 4  wff  ( z  e.  y  <->  E. w
( w  e.  x  /\  A. y ph )
)
1716, 4wal 1549 . . 3  wff  A. z
( z  e.  y  <->  E. w ( w  e.  x  /\  A. y ph ) )
1817, 2wex 1550 . 2  wff  E. y A. z ( z  e.  y  <->  E. w ( w  e.  x  /\  A. y ph ) )
1910, 18wi 4 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  4313  axnulALT  4328
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