Theorem reuuniss2 2881

Description: Restriction of a unique element to a smaller class.

Assertion

Ref	Expression
reuuniss2	|- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. A | ph} = U.{x e. B | ps})

Distinct variable groups:   x,A   x,B

Proof of Theorem reuuniss2

Step	Hyp	Ref	Expression
1		reuuni4 2877	. . . . 5 |- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)
2		reucl 2875	. . . . . 6 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
3		ra4sbc 1987	. . . . . . 7 |- (U.{x e. A | ph} e. A -> (A.x e. A (ph -> ps) -> [U.{x e. A | ph} / x](ph -> ps)))
4		sbcimg 1960	. . . . . . 7 |- (U.{x e. A | ph} e. A -> ([U.{x e. A | ph} / x](ph -> ps) <-> ([U.{x e. A | ph} / x]ph -> [U.{x e. A | ph} / x]ps)))
5	3, 4	sylibd 202	. . . . . 6 |- (U.{x e. A | ph} e. A -> (A.x e. A (ph -> ps) -> ([U.{x e. A | ph} / x]ph -> [U.{x e. A | ph} / x]ps)))
6	2, 5	syl 10	. . . . 5 |- (E!x e. A ph -> (A.x e. A (ph -> ps) -> ([U.{x e. A | ph} / x]ph -> [U.{x e. A | ph} / x]ps)))
7	1, 6	mpid 47	. . . 4 |- (E!x e. A ph -> (A.x e. A (ph -> ps) -> [U.{x e. A | ph} / x]ps))
8		reuss2 2265	. . . 4 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. A ph)
9		simplr 413	. . . 4 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> A.x e. A (ph -> ps))
10	7, 8, 9	sylc 68	. . 3 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> [U.{x e. A | ph} / x]ps)
11		hbrab1 1764	. . . . . 6 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
12	11	hbuni 2499	. . . . 5 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
13	12	hbsbc1g 1938	. . . . 5 |- (U.{x e. A | ph} e. B -> ([U.{x e. A | ph} / x]ps -> A.x[U.{x e. A | ph} / x]ps))
14		sbceq1a 1934	. . . . 5 |- (x = U.{x e. A | ph} -> (ps <-> [U.{x e. A | ph} / x]ps))
15	12, 13, 14	reuuni2f 2873	. . . 4 |- ((U.{x e. A | ph} e. B /\ E!x e. B ps) -> ([U.{x e. A | ph} / x]ps <-> U.{x e. B | ps} = U.{x e. A | ph}))
16	8, 2	syl 10	. . . . 5 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. A | ph} e. A)
17		ssel 2053	. . . . . 6 |- (A (_ B -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
18	17	ad2antrr 404	. . . . 5 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
19	16, 18	mpd 26	. . . 4 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. A | ph} e. B)
20		simprr 415	. . . 4 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. B ps)
21	15, 19, 20	sylanc 471	. . 3 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ([U.{x e. A | ph} / x]ps <-> U.{x e. B | ps} = U.{x e. A | ph}))
22	10, 21	mpbid 195	. 2 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. B | ps} = U.{x e. A | ph})
23	22	eqcomd 1472	1 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. A | ph} = U.{x e. B | ps})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  [wsbc 1166  A.wral 1637  E.wrex 1638  E!wreu 1639  {crab 1640   (_ wss 2037  U.cuni 2493

This theorem is referenced by:  grpidinv2 7994  grpinv 8003

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-uni 2494

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