Theorem reuuniss 3815

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

Assertion

Ref	Expression
reuuniss	|- ((A C_ 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 2871	. . . 4 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)
2		reuuni4 3813	. . . 4 |- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)
3	1, 2	syl 12	. . 3 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> [U.{x e. A | ph} / x]ph)
4		reucl 3213	. . . . . 6 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
5	1, 4	syl 12	. . . . 5 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> U.{x e. A | ph} e. A)
6		ssel 2615	. . . . . 6 |- (A C_ B -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
7	6	3ad2ant1 897	. . . . 5 |- ((A C_ 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))
8	5, 7	mpd 29	. . . 4 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> U.{x e. A | ph} e. B)
9		simp3 878	. . . 4 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. B ph)
10		hbrab1 2257	. . . . . 6 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
11	10	hbuni 3183	. . . . 5 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
12	11	hbsbc1g 2461	. . . . 5 |- (U.{x e. A | ph} e. B -> ([U.{x e. A | ph} / x]ph -> A.x[U.{x e. A | ph} / x]ph))
13		sbceq1a 2456	. . . . 5 |- (x = U.{x e. A | ph} -> (ph <-> [U.{x e. A | ph} / x]ph))
14	11, 12, 13	reuuni2f 3810	. . . 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}))
15	8, 9, 14	syl11anc 524	. . 3 |- ((A C_ 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 212	. 2 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> U.{x e. B | ph} = U.{x e. A | ph})
17	16	eqcomd 1889	1 |- ((A C_ 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 163   /\ w3a 858   = wceq 1298   e. wcel 1300  [wsbc 1534  E.wrex 2106  E!wreu 2107  {crab 2108   C_ wss 2593  U.cuni 3177

This theorem is referenced by:  mouniss 3816  supxrre 7292

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-uni 3178

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