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Theorem repizf2lem 3942
Description: Lemma for repizf2 3943. If we have a function-like proposition which provides at most one value of  y for each  x in a set  w, we can change "at most one" to "exactly one" by restricting the values of  x to those values for which the proposition provides a value of  y. (Contributed by Jim Kingdon, 7-Sep-2018.)
Assertion
Ref Expression
repizf2lem  |-  ( A. x  e.  w  E* y ph  <->  A. x  e.  {
x  e.  w  |  E. y ph } E! y ph )

Proof of Theorem repizf2lem
StepHypRef Expression
1 df-mo 1920 . . . 4  |-  ( E* y ph  <->  ( E. y ph  ->  E! y ph ) )
21imbi2i 219 . . 3  |-  ( ( x  e.  w  ->  E* y ph )  <->  ( x  e.  w  ->  ( E. y ph  ->  E! y ph ) ) )
32albii 1375 . 2  |-  ( A. x ( x  e.  w  ->  E* y ph )  <->  A. x ( x  e.  w  ->  ( E. y ph  ->  E! y ph ) ) )
4 df-ral 2328 . 2  |-  ( A. x  e.  w  E* y ph  <->  A. x ( x  e.  w  ->  E* y ph ) )
5 df-ral 2328 . . 3  |-  ( A. x  e.  { x  e.  w  |  E. y ph } E! y
ph 
<-> 
A. x ( x  e.  { x  e.  w  |  E. y ph }  ->  E! y ph ) )
6 rabid 2502 . . . . . 6  |-  ( x  e.  { x  e.  w  |  E. y ph }  <->  ( x  e.  w  /\  E. y ph ) )
76imbi1i 231 . . . . 5  |-  ( ( x  e.  { x  e.  w  |  E. y ph }  ->  E! y ph )  <->  ( (
x  e.  w  /\  E. y ph )  ->  E! y ph ) )
8 impexp 254 . . . . 5  |-  ( ( ( x  e.  w  /\  E. y ph )  ->  E! y ph )  <->  ( x  e.  w  -> 
( E. y ph  ->  E! y ph )
) )
97, 8bitri 177 . . . 4  |-  ( ( x  e.  { x  e.  w  |  E. y ph }  ->  E! y ph )  <->  ( x  e.  w  ->  ( E. y ph  ->  E! y ph ) ) )
109albii 1375 . . 3  |-  ( A. x ( x  e. 
{ x  e.  w  |  E. y ph }  ->  E! y ph )  <->  A. x ( x  e.  w  ->  ( E. y ph  ->  E! y ph ) ) )
115, 10bitri 177 . 2  |-  ( A. x  e.  { x  e.  w  |  E. y ph } E! y
ph 
<-> 
A. x ( x  e.  w  ->  ( E. y ph  ->  E! y ph ) ) )
123, 4, 113bitr4i 205 1  |-  ( A. x  e.  w  E* y ph  <->  A. x  e.  {
x  e.  w  |  E. y ph } E! y ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257   E.wex 1397    e. wcel 1409   E!weu 1916   E*wmo 1917   A.wral 2323   {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-sb 1662  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-rab 2332
This theorem is referenced by:  repizf2  3943
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