Theorem reuuni2f 2879

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 1945 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-17 970	. . . . . 6 |- (y e. A -> A.x y e. A)
3	1, 2	hbel 1564	. . . . 5 |- (B e. A -> A.x B e. A)
4		hbreu1 1766	. . . . . . 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 1770	. . . . . . . . . 10 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
8	7	hbuni 2505	. . . . . . . . 9 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
9	8, 1	hbeq 1563	. . . . . . . 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 1111	. . . . . 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 1109	. . . . 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 1102	. . . 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 1532	. . . . 5 |- (x = B -> (x e. A <-> B e. A))
15		reuuni2f.3	. . . . . . 7 |- (x = B -> (ph <-> ps))
16		eqeq2 1482	. . . . . . 7 |- (x = B -> (U.{x e. A | ph} = x <-> U.{x e. A | ph} = B))
17	15, 16	bibi12d 628	. . . . . 6 |- (x = B -> ((ph <-> U.{x e. A | ph} = x) <-> (ps <-> U.{x e. A | ph} = B)))
18	17	imbi2d 611	. . . . 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 625	. . . 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 2878	. . . . 5 |- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))
21	20	ex 373	. . . 4 |- (x e. A -> (E!x e. A ph -> (ph <-> U.{x e. A | ph} = x)))
22	1, 13, 19, 21	vtoclgf 1843	. . 3 |- (B e. A -> (B e. A -> (E!x e. A ph -> (ps <-> U.{x e. A | ph} = B))))
23	22	pm2.43i 64	. 2 |- (B e. A -> (E!x e. A ph -> (ps <-> U.{x e. A | ph} = B)))
24	23	imp 350	1 |- ((B e. A /\ E!x e. A ph) -> (ps <-> U.{x e. A | ph} = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E!wreu 1645  {crab 1646  U.cuni 2499

This theorem is referenced by:  reuuni2 2880  reuuniss 2885  reuuniss2 2887  reuunixfr 2902  minvecdist 8544

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-reu 1649  df-rab 1650  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-uni 2500

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