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Axiom ax-rep 4131
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 5330). 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 1603. Another common variant is derived as axrep5 4136, where you can find some further remarks. A slightly more compact version is shown as axrep2 4133. A quite different variant is zfrep6 5748, which if used in place of ax-rep 4131 would also require that the Separation Scheme axsep 4140 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 7561 and the Boundedness Axiom bnd 7562.

Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 4140, Null Set axnul 4148, and Pairing axpr 4213, 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 4141, ax-nul 4149, and ax-pr 4214 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 1527 . . . . . 6  wff  A. y ph
4 vz . . . . . . 7  set  z
54, 2weq 1624 . . . . . 6  wff  z  =  y
63, 5wi 4 . . . . 5  wff  ( A. y ph  ->  z  =  y )
76, 4wal 1527 . . . 4  wff  A. z
( A. y ph  ->  z  =  y )
87, 2wex 1528 . . 3  wff  E. y A. z ( A. y ph  ->  z  =  y )
9 vw . . 3  set  w
108, 9wal 1527 . 2  wff  A. w E. y A. z ( A. y ph  ->  z  =  y )
114, 2wel 1685 . . . . 5  wff  z  e.  y
12 vx . . . . . . . 8  set  x
139, 12wel 1685 . . . . . . 7  wff  w  e.  x
1413, 3wa 358 . . . . . 6  wff  ( w  e.  x  /\  A. y ph )
1514, 9wex 1528 . . . . 5  wff  E. w
( w  e.  x  /\  A. y ph )
1611, 15wb 176 . . . 4  wff  ( z  e.  y  <->  E. w
( w  e.  x  /\  A. y ph )
)
1716, 4wal 1527 . . 3  wff  A. z
( z  e.  y  <->  E. w ( w  e.  x  /\  A. y ph ) )
1817, 2wex 1528 . 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  4132  axnulALT  4147
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