Theorem reuuniss 2889

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 2276	. . . 4 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)
2		reuuni4 2887	. . . 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 1772	. . . . . 6 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
5	4	hbuni 2509	. . . . 5 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
6	5	hbsbc1g 1948	. . . . 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 1944	. . . . 5 |- (x = U.{x e. A | ph} -> (ph <-> [U.{x e. A | ph} / x]ph))
8	5, 6, 7	reuuni2f 2883	. . . 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 2885	. . . . . 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 2063	. . . . . 6 |- (A (_ B -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
12	11	3ad2ant1 800	. . . . 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 790	. . . 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 1480	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 775   = wceq 956   e. wcel 958  [wsbc 1170  E.wrex 1646  E!wreu 1647  {crab 1648   (_ wss 2047  U.cuni 2503

This theorem is referenced by:  mouniss 2890  supxrre 6083

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-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  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

Copyright terms: Public domain