Theorem reuuniss 2905

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 2287	. . . 4 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)
2		reuuni4 2903	. . . 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 1779	. . . . . 6 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
5	4	hbuni 2523	. . . . 5 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
6	5	hbsbc1g 1955	. . . . 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 1951	. . . . 5 |- (x = U.{x e. A | ph} -> (ph <-> [U.{x e. A | ph} / x]ph))
8	5, 6, 7	reuuni2f 2899	. . . 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 2901	. . . . . 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 2074	. . . . . 6 |- (A (_ B -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
12	11	3ad2ant1 804	. . . . 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 794	. . . 4 |- ((A (_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. B ph)
15	8, 13, 14	sylanc 474	. . 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 1487	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 779   = wceq 960   e. wcel 962  [wsbc 1174  E.wrex 1653  E!wreu 1654  {crab 1655   (_ wss 2058  U.cuni 2517

This theorem is referenced by:  mouniss 2906  supxrre 6115

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-sep 2718  ax-pow 2758  ax-un 2882

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-reu 1658  df-rab 1659  df-v 1819  df-sbc 1949  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-uni 2518

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