Theorem cnsubsp 15426

Description: Continuity of a restriction from a subspace.

Hypothesis

Ref	Expression
cnsubsp.1	|- X = U.J

Assertion

Ref	Expression
cnsubsp	|- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> (F |` A) e. ((subSp` <.A, J>.) Cn K))

Proof of Theorem cnsubsp

Step	Hyp	Ref	Expression
1		stoig3 10253	. . . . 5 |- ((J e. Top /\ A C_ U.J) -> (subSp` <.A, J>.) e. Top)
2		cnsubsp.1	. . . . . 6 |- X = U.J
3	2	sseq2i 2642	. . . . 5 |- (A C_ X <-> A C_ U.J)
4	1, 3	sylan2b 501	. . . 4 |- ((J e. Top /\ A C_ X) -> (subSp` <.A, J>.) e. Top)
5	4	ad2ant2r 445	. . 3 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> (subSp` <.A, J>.) e. Top)
6		simplr 449	. . 3 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> K e. Top)
7		eqid 1884	. . . 4 |- U.(subSp` <.A, J>.) = U.(subSp` <.A, J>.)
8		eqid 1884	. . . 4 |- U.K = U.K
9	7, 8	iscn 9034	. . 3 |- (((subSp` <.A, J>.) e. Top /\ K e. Top) -> ((F |` A) e. ((subSp` <.A, J>.) Cn K) <-> ((F |` A):U.(subSp` <.A, J>.)-->U.K /\ A.o e. K (`'(F |` A)"o) e. (subSp` <.A, J>.))))
10	5, 6, 9	syl11anc 524	. 2 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> ((F |` A) e. ((subSp` <.A, J>.) Cn K) <-> ((F |` A):U.(subSp` <.A, J>.)-->U.K /\ A.o e. K (`'(F |` A)"o) e. (subSp` <.A, J>.))))
11		df-f 4010	. . . . 5 |- ((F |` A):A-->U.K <-> ((F |` A) Fn A /\ ran ( F |` A) C_ U.K))
12		df-fn 4009	. . . . . 6 |- ((F |` A) Fn A <-> (Fun (F |` A) /\ dom ( F |` A) = A))
13	12	anbi1i 539	. . . . 5 |- (((F |` A) Fn A /\ ran ( F |` A) C_ U.K) <-> ((Fun (F |` A) /\ dom ( F |` A) = A) /\ ran ( F |` A) C_ U.K))
14	11, 13	bitri 190	. . . 4 |- ((F |` A):A-->U.K <-> ((Fun (F |` A) /\ dom ( F |` A) = A) /\ ran ( F |` A) C_ U.K))
15	2, 8	cnf 9038	. . . . . . . 8 |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> F:X-->U.K)
16		ffun 4565	. . . . . . . 8 |- (F:X-->U.K -> Fun F)
17		funres 4459	. . . . . . . 8 |- (Fun F -> Fun (F |` A))
18	15, 16, 17	3syl 24	. . . . . . 7 |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> Fun (F |` A))
19	18	3expa 1067	. . . . . 6 |- (((J e. Top /\ K e. Top) /\ F e. (J Cn K)) -> Fun (F |` A))
20	19	adantrl 430	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> Fun (F |` A))
21		simprl 450	. . . . . . 7 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> A C_ X)
22	15	3expa 1067	. . . . . . . . 9 |- (((J e. Top /\ K e. Top) /\ F e. (J Cn K)) -> F:X-->U.K)
23		fdm 4567	. . . . . . . . 9 |- (F:X-->U.K -> dom F = X)
24	22, 23	syl 12	. . . . . . . 8 |- (((J e. Top /\ K e. Top) /\ F e. (J Cn K)) -> dom F = X)
25	24	adantrl 430	. . . . . . 7 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> dom F = X)
26	21, 25	sseqtr4d 2654	. . . . . 6 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> A C_ dom F)
27		ssdmres 4235	. . . . . 6 |- (A C_ dom F <-> dom ( F |` A) = A)
28	26, 27	sylib 215	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> dom ( F |` A) = A)
29	20, 28	jca 310	. . . 4 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> (Fun (F |` A) /\ dom ( F |` A) = A))
30		resss 4237	. . . . . . 7 |- (F |` A) C_ F
31		rnss 4189	. . . . . . 7 |- ((F |` A) C_ F -> ran ( F |` A) C_ ran F)
32	30, 31	ax-mp 7	. . . . . 6 |- ran ( F |` A) C_ ran F
33	32	a1i 8	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> ran ( F |` A) C_ ran F)
34		frn 4569	. . . . . . . 8 |- (F:X-->U.K -> ran F C_ U.K)
35	15, 34	syl 12	. . . . . . 7 |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> ran F C_ U.K)
36	35	3expa 1067	. . . . . 6 |- (((J e. Top /\ K e. Top) /\ F e. (J Cn K)) -> ran F C_ U.K)
37	36	adantrl 430	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> ran F C_ U.K)
38	33, 37	sstrd 2627	. . . 4 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> ran ( F |` A) C_ U.K)
39	14, 29, 38	sylanbrc 527	. . 3 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> (F |` A):A-->U.K)
40		simpll 448	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> J e. Top)
41	3	biimpi 168	. . . . . 6 |- (A C_ X -> A C_ U.J)
42	41	ad2antrl 442	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> A C_ U.J)
43		stoig2 10252	. . . . 5 |- ((J e. Top /\ A C_ U.J) -> U.(subSp` <.A, J>.) = A)
44	40, 42, 43	syl11anc 524	. . . 4 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> U.(subSp` <.A, J>.) = A)
45	44	feq2d 4557	. . 3 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> ((F |` A):U.(subSp` <.A, J>.)-->U.K <-> (F |` A):A-->U.K))
46	39, 45	mpbird 213	. 2 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> (F |` A):U.(subSp` <.A, J>.)-->U.K)
47		simplll 452	. . . 4 |- ((((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) /\ o e. K) -> J e. Top)
48		simplrl 454	. . . 4 |- ((((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) /\ o e. K) -> A C_ X)
49		cnima 9044	. . . . . . . 8 |- (((J e. Top /\ K e. Top /\ F e. (J Cn K)) /\ o e. K) -> (`'F"o) e. J)
50	49	ex 402	. . . . . . 7 |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> (o e. K -> (`'F"o) e. J))
51	50	3expa 1067	. . . . . 6 |- (((J e. Top /\ K e. Top) /\ F e. (J Cn K)) -> (o e. K -> (`'F"o) e. J))
52	51	imp 377	. . . . 5 |- ((((J e. Top /\ K e. Top) /\ F e. (J Cn K)) /\ o e. K) -> (`'F"o) e. J)
53	52	adantlrl 434	. . . 4 |- ((((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) /\ o e. K) -> (`'F"o) e. J)
54		cnvresima 15359	. . . . 5 |- (`'(F |` A)"o) = ((`'F"o) i^i A)
55	54	a1i 8	. . . 4 |- ((((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) /\ o e. K) -> (`'(F |` A)"o) = ((`'F"o) i^i A))
56	2	elsubsp 10248	. . . 4 |- (((J e. Top /\ A C_ X) /\ ((`'F"o) e. J /\ (`'(F |` A)"o) = ((`'F"o) i^i A))) -> (`'(F |` A)"o) e. (subSp` <.A, J>.))
57	47, 48, 53, 55, 56	syl22anc 1101	. . 3 |- ((((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) /\ o e. K) -> (`'(F |` A)"o) e. (subSp` <.A, J>.))
58	57	r19.21aiva 2176	. 2 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> A.o e. K (`'(F |` A)"o) e. (subSp` <.A, J>.))
59	10, 46, 58	mpbir2and 802	1 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ F e. (J Cn K))) -> (F |` A) e. ((subSp` <.A, J>.) Cn K))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   i^i cin 2592   C_ wss 2593  <.cop 3046  U.cuni 3177  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857   Cn ccn 9028  subSpcsubsp 10242

This theorem is referenced by:  ivthALT 15454

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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695

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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-top 8861  df-topsp 8862  df-cn 9030  df-subsp 10243

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