Theorem reuuniss2 2891

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 2887	. . . . 5 |- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)
2		reucl 2885	. . . . . 6 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
3		ra4sbc 1997	. . . . . . 7 |- (U.{x e. A | ph} e. A -> (A.x e. A (ph -> ps) -> [U.{x e. A | ph} / x](ph -> ps)))
4		sbcimg 1970	. . . . . . 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 2275	. . . 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 413	. . . 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 1772	. . . . . 6 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
12	11	hbuni 2509	. . . . 5 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
13	12	hbsbc1g 1948	. . . . 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 1944	. . . . 5 |- (x = U.{x e. A | ph} -> (ps <-> [U.{x e. A | ph} / x]ps))
15	12, 13, 14	reuuni2f 2883	. . . 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 2063	. . . . . 6 |- (A (_ B -> (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. B))
18	17	ad2antrr 404	. . . . 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 415	. . . 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 471	. . 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 1480	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 956   e. wcel 958  [wsbc 1170  A.wral 1645  E.wrex 1646  E!wreu 1647  {crab 1648   (_ wss 2047  U.cuni 2503

This theorem is referenced by:  grpidinv2 8060  grpinv 8069

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-9 965  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

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