Theorem reuuniss 2884

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 2272	. . . 4 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)
2		reuuni4 2882	. . . 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 1769	. . . . . 6 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
5	4	hbuni 2504	. . . . 5 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
6	5	hbsbc1g 1944	. . . . 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 1940	. . . . 5 |- (x = U.{x e. A | ph} -> (ph <-> [U.{x e. A | ph} / x]ph))
8	5, 6, 7	reuuni2f 2878	. . . 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 2880	. . . . . 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 2059	. . . . . 6 |- (A (_ B -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
12	11	3ad2ant1 799	. . . . 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 789	. . . 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 1477	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 774   = wceq 954   e. wcel 956  [wsbc 1168  E.wrex 1643  E!wreu 1644  {crab 1645   (_ wss 2043  U.cuni 2498

This theorem is referenced by:  mouniss 2885  supxrre 6038

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-uni 2499

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