Theorem reiotass2 5111

Description: Restriction of a unique element to a smaller class. Compare reuuniss2 3817.

Assertion

Ref	Expression
reiotass2	|- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> (iotax(x e. A /\ ph)) = (iotax(x e. B /\ ps)))

Distinct variable groups:   x,A   x,B

Proof of Theorem reiotass2

Step	Hyp	Ref	Expression
1		reuss2 2870	. . . 4 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. A ph)
2		simplr 449	. . . 4 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> A.x e. A (ph -> ps))
3		reiota4 5107	. . . . 5 |- (E!x e. A ph -> [(iotax(x e. A /\ ph)) / x]ph)
4		reiotacl 5106	. . . . . 6 |- (E!x e. A ph -> (iotax(x e. A /\ ph)) e. A)
5		ra4sbc 2536	. . . . . . 7 |- ((iotax(x e. A /\ ph)) e. A -> (A.x e. A (ph -> ps) -> [(iotax(x e. A /\ ph)) / x](ph -> ps)))
6		sbcimg 2496	. . . . . . 7 |- ((iotax(x e. A /\ ph)) e. A -> ([(iotax(x e. A /\ ph)) / x](ph -> ps) <-> ([(iotax(x e. A /\ ph)) / x]ph -> [(iotax(x e. A /\ ph)) / x]ps)))
7	5, 6	sylibd 219	. . . . . 6 |- ((iotax(x e. A /\ ph)) e. A -> (A.x e. A (ph -> ps) -> ([(iotax(x e. A /\ ph)) / x]ph -> [(iotax(x e. A /\ ph)) / x]ps)))
8	4, 7	syl 12	. . . . 5 |- (E!x e. A ph -> (A.x e. A (ph -> ps) -> ([(iotax(x e. A /\ ph)) / x]ph -> [(iotax(x e. A /\ ph)) / x]ps)))
9	3, 8	mpid 58	. . . 4 |- (E!x e. A ph -> (A.x e. A (ph -> ps) -> [(iotax(x e. A /\ ph)) / x]ps))
10	1, 2, 9	sylc 83	. . 3 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> [(iotax(x e. A /\ ph)) / x]ps)
11	1, 4	syl 12	. . . . 5 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> (iotax(x e. A /\ ph)) e. A)
12		ssel 2615	. . . . . 6 |- (A C_ B -> ((iotax(x e. A /\ ph)) e. A -> (iotax(x e. A /\ ph)) e. B))
13	12	ad2antrr 440	. . . . 5 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ((iotax(x e. A /\ ph)) e. A -> (iotax(x e. A /\ ph)) e. B))
14	11, 13	mpd 29	. . . 4 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> (iotax(x e. A /\ ph)) e. B)
15		simprr 451	. . . 4 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. B ps)
16		hbiota1 5091	. . . . 5 |- (y e. (iotax(x e. A /\ ph)) -> A.x y e. (iotax(x e. A /\ ph)))
17	16	hbsbc1g 2461	. . . . 5 |- ((iotax(x e. A /\ ph)) e. B -> ([(iotax(x e. A /\ ph)) / x]ps -> A.x[(iotax(x e. A /\ ph)) / x]ps))
18		sbceq1a 2456	. . . . 5 |- (x = (iotax(x e. A /\ ph)) -> (ps <-> [(iotax(x e. A /\ ph)) / x]ps))
19	16, 17, 18	reiota2f 5109	. . . 4 |- (((iotax(x e. A /\ ph)) e. B /\ E!x e. B ps) -> ([(iotax(x e. A /\ ph)) / x]ps <-> (iotax(x e. B /\ ps)) = (iotax(x e. A /\ ph))))
20	14, 15, 19	syl11anc 524	. . 3 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ([(iotax(x e. A /\ ph)) / x]ps <-> (iotax(x e. B /\ ps)) = (iotax(x e. A /\ ph))))
21	10, 20	mpbid 212	. 2 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> (iotax(x e. B /\ ps)) = (iotax(x e. A /\ ph)))
22	21	eqcomd 1889	1 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> (iotax(x e. A /\ ph)) = (iotax(x e. B /\ ps)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  E.wrex 2106  E!wreu 2107   C_ wss 2593  iotacio 5087

This theorem is referenced by:  grpidinv2NEW 17119  grpinvNEW 17128

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  df-iota 5089

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