Theorem repizf2lem 4045

Description: Lemma for repizf2 4046. 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

Step	Hyp	Ref	Expression
1		df-mo 1979	. . . 4 |- ( E* y ph <-> ( E. y ph -> E! y ph ) )
2	1	imbi2i 225	. . 3 |- ( ( x e. w -> E* y ph ) <-> ( x e. w -> ( E. y ph -> E! y ph ) ) )
3	2	albii 1429	. 2 |- ( A. x ( x e. w -> E* y ph ) <-> A. x ( x e. w -> ( E. y ph -> E! y ph ) ) )
4		df-ral 2395	. 2 |- ( A. x e. w E* y ph <-> A. x ( x e. w -> E* y ph ) )
5		df-ral 2395	. . 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 2580	. . . . . 6 |- ( x e. { x e. w | E. y ph } <-> ( x e. w /\ E. y ph ) )
7	6	imbi1i 237	. . . . 5 |- ( ( x e. { x e. w | E. y ph } -> E! y ph ) <-> ( ( x e. w /\ E. y ph ) -> E! y ph ) )
8		impexp 261	. . . . 5 |- ( ( ( x e. w /\ E. y ph ) -> E! y ph ) <-> ( x e. w -> ( E. y ph -> E! y ph ) ) )
9	7, 8	bitri 183	. . . 4 |- ( ( x e. { x e. w | E. y ph } -> E! y ph ) <-> ( x e. w -> ( E. y ph -> E! y ph ) ) )
10	9	albii 1429	. . 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 ) ) )
11	5, 10	bitri 183	. 2 |- ( A. x e. { x e. w | E. y ph } E! y ph <-> A. x ( x e. w -> ( E. y ph -> E! y ph ) ) )
12	3, 4, 11	3bitr4i 211	1 |- ( A. x e. w E* y ph <-> A. x e. { x e. w | E. y ph } E! y ph )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1312   E.wex 1451   e. wcel 1463   E!weu 1975   E*wmo 1976   A.wral 2390   {crab 2394

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-sb 1719  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2395  df-rab 2399

This theorem is referenced by:  repizf2  4046