Theorem reuuni2f 2883

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 1948 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-17 971	. . . . . 6 |- (y e. A -> A.x y e. A)
3	1, 2	hbel 1566	. . . . 5 |- (B e. A -> A.x B e. A)
4		hbreu1 1768	. . . . . . 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 1772	. . . . . . . . . 10 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
8	7	hbuni 2509	. . . . . . . . 9 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
9	8, 1	hbeq 1565	. . . . . . . 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 1112	. . . . . 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 1110	. . . . 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 1103	. . . 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 1534	. . . . 5 |- (x = B -> (x e. A <-> B e. A))
15		reuuni2f.3	. . . . . . 7 |- (x = B -> (ph <-> ps))
16		eqeq2 1484	. . . . . . 7 |- (x = B -> (U.{x e. A | ph} = x <-> U.{x e. A | ph} = B))
17	15, 16	bibi12d 629	. . . . . 6 |- (x = B -> ((ph <-> U.{x e. A | ph} = x) <-> (ps <-> U.{x e. A | ph} = B)))
18	17	imbi2d 612	. . . . 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 626	. . . 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 2882	. . . . 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 1846	. . 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 954   = wceq 956   e. wcel 958  E!wreu 1647  {crab 1648  U.cuni 2503

This theorem is referenced by:  reuuni2 2884  reuuniss 2889  reuuniss2 2891  reuunixfr 2906  minvecdist 8585

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-reu 1651  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504

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