Theorem reuuniss2 2897

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

Assertion

Ref	Expression
reuuniss2	|- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. A | ph} = U.{x e. B | ps})

Distinct variable groups:   x,A   x,B

Proof of Theorem reuuniss2

Step	Hyp	Ref	Expression
1		reuuni4 2893	. . . . 5 |- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)
2		reucl 2891	. . . . . 6 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
3		ra4sbc 2000	. . . . . . 7 |- (U.{x e. A | ph} e. A -> (A.x e. A (ph -> ps) -> [U.{x e. A | ph} / x](ph -> ps)))
4		sbcimg 1973	. . . . . . 7 |- (U.{x e. A | ph} e. A -> ([U.{x e. A | ph} / x](ph -> ps) <-> ([U.{x e. A | ph} / x]ph -> [U.{x e. A | ph} / x]ps)))
5	3, 4	sylibd 202	. . . . . 6 |- (U.{x e. A | ph} e. A -> (A.x e. A (ph -> ps) -> ([U.{x e. A | ph} / x]ph -> [U.{x e. A | ph} / x]ps)))
6	2, 5	syl 10	. . . . 5 |- (E!x e. A ph -> (A.x e. A (ph -> ps) -> ([U.{x e. A | ph} / x]ph -> [U.{x e. A | ph} / x]ps)))
7	1, 6	mpid 47	. . . 4 |- (E!x e. A ph -> (A.x e. A (ph -> ps) -> [U.{x e. A | ph} / x]ps))
8		reuss2 2278	. . . 4 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. A ph)
9		simplr 415	. . . 4 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> A.x e. A (ph -> ps))
10	7, 8, 9	sylc 68	. . 3 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> [U.{x e. A | ph} / x]ps)
11		hbrab1 1775	. . . . . 6 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
12	11	hbuni 2513	. . . . 5 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
13	12	hbsbc1g 1951	. . . . 5 |- (U.{x e. A | ph} e. B -> ([U.{x e. A | ph} / x]ps -> A.x[U.{x e. A | ph} / x]ps))
14		sbceq1a 1947	. . . . 5 |- (x = U.{x e. A | ph} -> (ps <-> [U.{x e. A | ph} / x]ps))
15	12, 13, 14	reuuni2f 2889	. . . 4 |- ((U.{x e. A | ph} e. B /\ E!x e. B ps) -> ([U.{x e. A | ph} / x]ps <-> U.{x e. B | ps} = U.{x e. A | ph}))
16	8, 2	syl 10	. . . . 5 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. A | ph} e. A)
17		ssel 2066	. . . . . 6 |- (A (_ B -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
18	17	ad2antrr 406	. . . . 5 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
19	16, 18	mpd 26	. . . 4 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. A | ph} e. B)
20		simprr 417	. . . 4 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. B ps)
21	15, 19, 20	sylanc 473	. . 3 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ([U.{x e. A | ph} / x]ps <-> U.{x e. B | ps} = U.{x e. A | ph}))
22	10, 21	mpbid 195	. 2 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. B | ps} = U.{x e. A | ph})
23	22	eqcomd 1483	1 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> U.{x e. A | ph} = U.{x e. B | ps})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  [wsbc 1172  A.wral 1648  E.wrex 1649  E!wreu 1650  {crab 1651   (_ wss 2050  U.cuni 2507

This theorem is referenced by:  grpidinv2 8056  grpinv 8065

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508

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