Theorem reuuni2f 3810

Description: U.{x e. A | ph} is an explicit representation of "the unique element in A such that ph." This theorem shows a condition that allows us to represent this element with a class expression B. The second hypothesis is a weakened bound variable condition that allows hbsbc1g 2461 to be used.

Hypotheses

Ref	Expression
reuuni2f.1	|- (y e. B -> A.x y e. B)
reuuni2f.2	|- (B e. A -> (ps -> A.xps))
reuuni2f.3	|- (x = B -> (ph <-> ps))

Assertion

Ref	Expression
reuuni2f	|- ((B e. A /\ E!x e. A ph) -> (ps <-> U.{x e. A | ph} = B))

Distinct variable groups:   ph,y   x,y,A   y,B

Proof of Theorem reuuni2f

Step	Hyp	Ref	Expression
1		reuuni2f.1	. . . 4 |- (y e. B -> A.x y e. B)
2		ax-17 1317	. . . . . 6 |- (y e. A -> A.x y e. A)
3	1, 2	hbel 1996	. . . . 5 |- (B e. A -> A.x B e. A)
4		hbreu1 2252	. . . . . . 7 |- (E!x e. A ph -> A.xE!x e. A ph)
5	4	a1i 8	. . . . . 6 |- (B e. A -> (E!x e. A ph -> A.xE!x e. A ph))
6		reuuni2f.2	. . . . . . 7 |- (B e. A -> (ps -> A.xps))
7		hbrab1 2257	. . . . . . . . . 10 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
8	7	hbuni 3183	. . . . . . . . 9 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
9	8, 1	hbeq 1995	. . . . . . . 8 |- (U.{x e. A | ph} = B -> A.xU.{x e. A | ph} = B)
10	9	a1i 8	. . . . . . 7 |- (B e. A -> (U.{x e. A | ph} = B -> A.xU.{x e. A | ph} = B))
11	3, 6, 10	hbbid 1470	. . . . . 6 |- (B e. A -> ((ps <-> U.{x e. A | ph} = B) -> A.x(ps <-> U.{x e. A | ph} = B)))
12	3, 5, 11	hbimd 1468	. . . . 5 |- (B e. A -> ((E!x e. A ph -> (ps <-> U.{x e. A | ph} = B)) -> A.x(E!x e. A ph -> (ps <-> U.{x e. A | ph} = B))))
13	3, 12	hbim1 1458	. . . 4 |- ((B e. A -> (E!x e. A ph -> (ps <-> U.{x e. A | ph} = B))) -> A.x(B e. A -> (E!x e. A ph -> (ps <-> U.{x e. A | ph} = B))))
14		eleq1 1957	. . . . 5 |- (x = B -> (x e. A <-> B e. A))
15		reuuni2f.3	. . . . . . 7 |- (x = B -> (ph <-> ps))
16		eqeq2 1893	. . . . . . 7 |- (x = B -> (U.{x e. A | ph} = x <-> U.{x e. A | ph} = B))
17	15, 16	bibi12d 691	. . . . . 6 |- (x = B -> ((ph <-> U.{x e. A | ph} = x) <-> (ps <-> U.{x e. A | ph} = B)))
18	17	imbi2d 674	. . . . 5 |- (x = B -> ((E!x e. A ph -> (ph <-> U.{x e. A | ph} = x)) <-> (E!x e. A ph -> (ps <-> U.{x e. A | ph} = B))))
19	14, 18	imbi12d 688	. . . 4 |- (x = B -> ((x e. A -> (E!x e. A ph -> (ph <-> U.{x e. A | ph} = x))) <-> (B e. A -> (E!x e. A ph -> (ps <-> U.{x e. A | ph} = B)))))
20		reuuni1 3808	. . . . 5 |- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))
21	20	ex 402	. . . 4 |- (x e. A -> (E!x e. A ph -> (ph <-> U.{x e. A | ph} = x)))
22	1, 13, 19, 21	vtoclgf 2345	. . 3 |- (B e. A -> (B e. A -> (E!x e. A ph -> (ps <-> U.{x e. A | ph} = B))))
23	22	pm2.43i 78	. 2 |- (B e. A -> (E!x e. A ph -> (ps <-> U.{x e. A | ph} = B)))
24	23	imp 377	1 |- ((B e. A /\ E!x e. A ph) -> (ps <-> U.{x e. A | ph} = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E!wreu 2107  {crab 2108  U.cuni 3177

This theorem is referenced by:  reuuni2 3811  reuuniss 3815  reuuniss2 3817  reuunixfr 3850  minvecdist 9930

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-uni 3178

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