Theorem repizf2lem 3905

Description: Lemma for repizf2 3906. 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*yph <-> A.x e. {x e. w | E.yph }E!yph)

Proof of Theorem repizf2lem

Step	Hyp	Ref	Expression
1		df-mo 1901	. . . 4 |- (E*yph <-> (E.yph -> E!yph))
2	1	imbi2i 215	. . 3 |- ((x e. w -> E*yph) <-> (x e. w -> (E.yph -> E!yph)))
3	2	albii 1356	. 2 |- (A.x(x e. w -> E*yph) <-> A.x(x e. w -> (E.yph -> E!yph)))
4		df-ral 2305	. 2 |- (A.x e. w E*yph <-> A.x(x e. w -> E*yph))
5		df-ral 2305	. . 3 |- (A.x e. {x e. w | E.yph }E!yph <-> A.x(x e. {x e. w | E.yph } -> E!yph))
6		rabid 2479	. . . . . 6 |- (x e. {x e. w | E.yph } <-> (x e. w /\ E.yph))
7	6	imbi1i 227	. . . . 5 |- ((x e. {x e. w | E.yph } -> E!yph) <-> ((x e. w /\ E.yph) -> E!yph))
8		impexp 250	. . . . 5 |- (((x e. w /\ E.yph) -> E!yph) <-> (x e. w -> (E.yph -> E!yph)))
9	7, 8	bitri 173	. . . 4 |- ((x e. {x e. w | E.yph } -> E!yph) <-> (x e. w -> (E.yph -> E!yph)))
10	9	albii 1356	. . 3 |- (A.x(x e. {x e. w | E.yph } -> E!yph) <-> A.x(x e. w -> (E.yph -> E!yph)))
11	5, 10	bitri 173	. 2 |- (A.x e. {x e. w | E.yph }E!yph <-> A.x(x e. w -> (E.yph -> E!yph)))
12	3, 4, 11	3bitr4i 201	1 |- (A.x e. w E*yph <-> A.x e. {x e. w | E.yph }E!yph)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378   e. wcel 1390  E!weu 1897  E*wmo 1898  A.wral 2300   {crab 2304

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-sb 1643  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rab 2309

This theorem is referenced by:  repizf2  3906