Theorem cnres2 15890

Description: The restriction of a continuous function to a subset is continuous.

Hypotheses

Ref	Expression
cnres2.1	|- X = U.J
cnres2.2	|- Y = U.K

Assertion

Ref	Expression
cnres2	|- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> (F |` A) e. ((subSp` <.A, J>.) Cn (subSp` <.B, K>.)))

Distinct variable groups:   x,J   x,K   x,F   x,X   x,Y   x,A   x,B

Proof of Theorem cnres2

Step	Hyp	Ref	Expression
1		stoig3 10253	. . . . 5 |- ((J e. Top /\ A C_ U.J) -> (subSp` <.A, J>.) e. Top)
2		cnres2.1	. . . . . 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 /\ B C_ Y)) -> (subSp` <.A, J>.) e. Top)
6	5	3adant3 896	. 2 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> (subSp` <.A, J>.) e. Top)
7		simp1r 901	. 2 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> K e. Top)
8		simp1l 900	. . 3 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> J e. Top)
9		simp3l 904	. . 3 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> F e. (J Cn K))
10		simp2l 902	. . 3 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> A C_ X)
11	2	cnres 15889	. . 3 |- (((J e. Top /\ K e. Top) /\ (F e. (J Cn K) /\ A C_ X)) -> (F |` A) e. ((subSp` <.A, J>.) Cn K))
12	8, 7, 9, 10, 11	syl22anc 1101	. 2 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> (F |` A) e. ((subSp` <.A, J>.) Cn K))
13		simp2r 903	. 2 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> B C_ Y)
14		stoig2 10252	. . . . . . . . . 10 |- ((J e. Top /\ A C_ U.J) -> U.(subSp` <.A, J>.) = A)
15	14, 3	sylan2b 501	. . . . . . . . 9 |- ((J e. Top /\ A C_ X) -> U.(subSp` <.A, J>.) = A)
16	15	ad2ant2r 445	. . . . . . . 8 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y)) -> U.(subSp` <.A, J>.) = A)
17	16	adantr 425	. . . . . . 7 |- ((((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y)) /\ F e. (J Cn K)) -> U.(subSp` <.A, J>.) = A)
18	17	raleqdv 2269	. . . . . 6 |- ((((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y)) /\ F e. (J Cn K)) -> (A.x e. U.(subSp` <.A, J>.)((F |` A)` x) e. B <-> A.x e. A ((F |` A)` x) e. B))
19		fvres 4691	. . . . . . . 8 |- (x e. A -> ((F |` A)` x) = (F` x))
20	19	eleq1d 1963	. . . . . . 7 |- (x e. A -> (((F |` A)` x) e. B <-> (F` x) e. B))
21	20	ralbiia 2133	. . . . . 6 |- (A.x e. A ((F |` A)` x) e. B <-> A.x e. A (F` x) e. B)
22	18, 21	syl6rbb 596	. . . . 5 |- ((((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y)) /\ F e. (J Cn K)) -> (A.x e. A (F` x) e. B <-> A.x e. U.(subSp` <.A, J>.)((F |` A)` x) e. B))
23	22	biimpa 460	. . . 4 |- (((((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y)) /\ F e. (J Cn K)) /\ A.x e. A (F` x) e. B) -> A.x e. U.(subSp` <.A, J>.)((F |` A)` x) e. B)
24	23	anasss 488	. . 3 |- ((((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y)) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> A.x e. U.(subSp` <.A, J>.)((F |` A)` x) e. B)
25	24	3impa 1062	. 2 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> A.x e. U.(subSp` <.A, J>.)((F |` A)` x) e. B)
26		eqid 1884	. . 3 |- U.(subSp` <.A, J>.) = U.(subSp` <.A, J>.)
27		cnres2.2	. . 3 |- Y = U.K
28	26, 27	cnimass 15888	. 2 |- ((((subSp` <.A, J>.) e. Top /\ K e. Top /\ (F |` A) e. ((subSp` <.A, J>.) Cn K)) /\ (B C_ Y /\ A.x e. U.(subSp` <.A, J>.)((F |` A)` x) e. B)) -> (F |` A) e. ((subSp` <.A, J>.) Cn (subSp` <.B, K>.)))
29	6, 7, 12, 13, 25, 28	syl32anc 1108	1 |- (((J e. Top /\ K e. Top) /\ (A C_ X /\ B C_ Y) /\ (F e. (J Cn K) /\ A.x e. A (F` x) e. B)) -> (F |` A) e. ((subSp` <.A, J>.) Cn (subSp` <.B, K>.)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  <.cop 3046  U.cuni 3177   |` cres 3988  ` cfv 3998  (class class class)co 4884  Topctop 8857   Cn ccn 9028  subSpcsubsp 10242

This theorem is referenced by:  cnresoprab 15915

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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