Theorem repizf2lem 4160

Description: Lemma for repizf2 4161. 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 2030	. . . 4 |- ( E* y ph <-> ( E. y ph -> E! y ph ) )
2	1	imbi2i 226	. . 3 |- ( ( x e. w -> E* y ph ) <-> ( x e. w -> ( E. y ph -> E! y ph ) ) )
3	2	albii 1470	. 2 |- ( A. x ( x e. w -> E* y ph ) <-> A. x ( x e. w -> ( E. y ph -> E! y ph ) ) )
4		df-ral 2460	. 2 |- ( A. x e. w E* y ph <-> A. x ( x e. w -> E* y ph ) )
5		df-ral 2460	. . 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 2652	. . . . . 6 |- ( x e. { x e. w | E. y ph } <-> ( x e. w /\ E. y ph ) )
7	6	imbi1i 238	. . . . 5 |- ( ( x e. { x e. w | E. y ph } -> E! y ph ) <-> ( ( x e. w /\ E. y ph ) -> E! y ph ) )
8		impexp 263	. . . . 5 |- ( ( ( x e. w /\ E. y ph ) -> E! y ph ) <-> ( x e. w -> ( E. y ph -> E! y ph ) ) )
9	7, 8	bitri 184	. . . 4 |- ( ( x e. { x e. w | E. y ph } -> E! y ph ) <-> ( x e. w -> ( E. y ph -> E! y ph ) ) )
10	9	albii 1470	. . 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 184	. 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 212	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 104   <-> wb 105   A.wal 1351   E.wex 1492   E!weu 2026   E*wmo 2027   e. wcel 2148   A.wral 2455   {crab 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-sb 1763  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-rab 2464

This theorem is referenced by:  repizf2  4161