Theorem reuuniss 2879

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

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem reuuniss

Step	Hyp	Ref	Expression
1		reuss 2266	. . . 4 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)
2		reuuni4 2877	. . . 4 |- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)
3	1, 2	syl 10	. . 3 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> [U.{x e. A | ph} / x]ph)
4		hbrab1 1764	. . . . . 6 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
5	4	hbuni 2499	. . . . 5 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
6	5	hbsbc1g 1938	. . . . 5 |- (U.{x e. A | ph} e. B -> ([U.{x e. A | ph} / x]ph -> A.x[U.{x e. A | ph} / x]ph))
7		sbceq1a 1934	. . . . 5 |- (x = U.{x e. A | ph} -> (ph <-> [U.{x e. A | ph} / x]ph))
8	5, 6, 7	reuuni2f 2873	. . . 4 |- ((U.{x e. A | ph} e. B /\ E!x e. B ph) -> ([U.{x e. A | ph} / x]ph <-> U.{x e. B | ph} = U.{x e. A | ph}))
9		reucl 2875	. . . . . 6 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
10	1, 9	syl 10	. . . . 5 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> U.{x e. A | ph} e. A)
11		ssel 2053	. . . . . 6 |- (A (_ B -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
12	11	3ad2ant1 798	. . . . 5 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
13	10, 12	mpd 26	. . . 4 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> U.{x e. A | ph} e. B)
14		3simp3 788	. . . 4 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. B ph)
15	8, 13, 14	sylanc 471	. . 3 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> ([U.{x e. A | ph} / x]ph <-> U.{x e. B | ph} = U.{x e. A | ph}))
16	3, 15	mpbid 195	. 2 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> U.{x e. B | ph} = U.{x e. A | ph})
17	16	eqcomd 1472	1 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> U.{x e. A | ph} = U.{x e. B | ph})

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 773   = wceq 953   e. wcel 955  [wsbc 1166  E.wrex 1638  E!wreu 1639  {crab 1640   (_ wss 2037  U.cuni 2493

This theorem is referenced by:  mouniss 2880  supxrre 6030

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-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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