Theorem cnextf 21870

Description: Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)

Hypotheses

Ref	Expression
cnextf.1	|- C = U. J
cnextf.2	|- B = U. K
cnextf.3	|- ( ph -> J e. Top )
cnextf.4	|- ( ph -> K e. Haus )
cnextf.5	|- ( ph -> F : A --> B )
cnextf.a	|- ( ph -> A C_ C )
cnextf.6	|- ( ph -> ( ( cls ` J ) ` A ) = C )
cnextf.7	|- ( ( ph /\ x e. C ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) =/= (/) )

Assertion

Ref	Expression
cnextf	|- ( ph -> ( ( JCnExt K ) ` F ) : C --> B )

Distinct variable groups:   x, A   x, B   x, C   x, F   x, J   x, K   ph, x

Proof of Theorem cnextf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnextf.3	. . . 4 |- ( ph -> J e. Top )
2		cnextf.4	. . . 4 |- ( ph -> K e. Haus )
3		cnextf.5	. . . 4 |- ( ph -> F : A --> B )
4		cnextf.a	. . . 4 |- ( ph -> A C_ C )
5		cnextf.1	. . . . 5 |- C = U. J
6		cnextf.2	. . . . 5 |- B = U. K
7	5, 6	cnextfun 21868	. . . 4 |- ( ( ( J e. Top /\ K e. Haus ) /\ ( F : A --> B /\ A C_ C ) ) -> Fun ( ( JCnExt K ) ` F ) )
8	1, 2, 3, 4, 7	syl22anc 1327	. . 3 |- ( ph -> Fun ( ( JCnExt K ) ` F ) )
9		simpl 473	. . . . . . 7 |- ( ( ph /\ x e. C ) -> ph )
10		cnextf.6	. . . . . . . . 9 |- ( ph -> ( ( cls ` J ) ` A ) = C )
11	10	eleq2d 2687	. . . . . . . 8 |- ( ph -> ( x e. ( ( cls ` J ) ` A ) <-> x e. C ) )
12	11	biimpar 502	. . . . . . 7 |- ( ( ph /\ x e. C ) -> x e. ( ( cls ` J ) ` A ) )
13		cnextf.7	. . . . . . . 8 |- ( ( ph /\ x e. C ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) =/= (/) )
14		n0 3931	. . . . . . . 8 |- ( ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) =/= (/) <-> E. y y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) )
15	13, 14	sylib 208	. . . . . . 7 |- ( ( ph /\ x e. C ) -> E. y y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) )
16		haustop 21135	. . . . . . . . . . . . . 14 |- ( K e. Haus -> K e. Top )
17	2, 16	syl 17	. . . . . . . . . . . . 13 |- ( ph -> K e. Top )
18	5, 6	cnextfval 21866	. . . . . . . . . . . . 13 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ C ) ) -> ( ( JCnExt K ) ` F ) = U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
19	1, 17, 3, 4, 18	syl22anc 1327	. . . . . . . . . . . 12 |- ( ph -> ( ( JCnExt K ) ` F ) = U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
20	19	eleq2d 2687	. . . . . . . . . . 11 |- ( ph -> ( <. x , y >. e. ( ( JCnExt K ) ` F ) <-> <. x , y >. e. U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) ) )
21		opeliunxp 5170	. . . . . . . . . . 11 |- ( <. x , y >. e. U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) <-> ( x e. ( ( cls ` J ) ` A ) /\ y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
22	20, 21	syl6bb 276	. . . . . . . . . 10 |- ( ph -> ( <. x , y >. e. ( ( JCnExt K ) ` F ) <-> ( x e. ( ( cls ` J ) ` A ) /\ y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) ) )
23	22	exbidv 1850	. . . . . . . . 9 |- ( ph -> ( E. y <. x , y >. e. ( ( JCnExt K ) ` F ) <-> E. y ( x e. ( ( cls ` J ) ` A ) /\ y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) ) )
24		19.42v 1918	. . . . . . . . 9 |- ( E. y ( x e. ( ( cls ` J ) ` A ) /\ y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) <-> ( x e. ( ( cls ` J ) ` A ) /\ E. y y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
25	23, 24	syl6bb 276	. . . . . . . 8 |- ( ph -> ( E. y <. x , y >. e. ( ( JCnExt K ) ` F ) <-> ( x e. ( ( cls ` J ) ` A ) /\ E. y y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) ) )
26	25	biimpar 502	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( cls ` J ) ` A ) /\ E. y y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) ) -> E. y <. x , y >. e. ( ( JCnExt K ) ` F ) )
27	9, 12, 15, 26	syl12anc 1324	. . . . . 6 |- ( ( ph /\ x e. C ) -> E. y <. x , y >. e. ( ( JCnExt K ) ` F ) )
28	25	simprbda 653	. . . . . . 7 |- ( ( ph /\ E. y <. x , y >. e. ( ( JCnExt K ) ` F ) ) -> x e. ( ( cls ` J ) ` A ) )
29	11	adantr 481	. . . . . . 7 |- ( ( ph /\ E. y <. x , y >. e. ( ( JCnExt K ) ` F ) ) -> ( x e. ( ( cls ` J ) ` A ) <-> x e. C ) )
30	28, 29	mpbid 222	. . . . . 6 |- ( ( ph /\ E. y <. x , y >. e. ( ( JCnExt K ) ` F ) ) -> x e. C )
31	27, 30	impbida 877	. . . . 5 |- ( ph -> ( x e. C <-> E. y <. x , y >. e. ( ( JCnExt K ) ` F ) ) )
32	31	abbi2dv 2742	. . . 4 |- ( ph -> C = { x | E. y <. x , y >. e. ( ( JCnExt K ) ` F ) } )
33		dfdm3 5310	. . . 4 |- dom ( ( JCnExt K ) ` F ) = { x | E. y <. x , y >. e. ( ( JCnExt K ) ` F ) }
34	32, 33	syl6reqr 2675	. . 3 |- ( ph -> dom ( ( JCnExt K ) ` F ) = C )
35		df-fn 5891	. . 3 |- ( ( ( JCnExt K ) ` F ) Fn C <-> ( Fun ( ( JCnExt K ) ` F ) /\ dom ( ( JCnExt K ) ` F ) = C ) )
36	8, 34, 35	sylanbrc 698	. 2 |- ( ph -> ( ( JCnExt K ) ` F ) Fn C )
37	19	rneqd 5353	. . 3 |- ( ph -> ran ( ( JCnExt K ) ` F ) = ran U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
38		rniun 5543	. . . 4 |- ran U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) = U_ x e. ( ( cls ` J ) ` A ) ran ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) )
39		vex 3203	. . . . . . . . 9 |- x e. _V
40	39	snnz 4309	. . . . . . . 8 |- { x } =/= (/)
41		rnxp 5564	. . . . . . . 8 |- ( { x } =/= (/) -> ran ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) = ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) )
42	40, 41	ax-mp 5	. . . . . . 7 |- ran ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) = ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F )
43	11	biimpa 501	. . . . . . . 8 |- ( ( ph /\ x e. ( ( cls ` J ) ` A ) ) -> x e. C )
44	6	toptopon 20722	. . . . . . . . . . 11 |- ( K e. Top <-> K e. (TopOn ` B ) )
45	17, 44	sylib 208	. . . . . . . . . 10 |- ( ph -> K e. (TopOn ` B ) )
46	45	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. C ) -> K e. (TopOn ` B ) )
47	5	toptopon 20722	. . . . . . . . . . . 12 |- ( J e. Top <-> J e. (TopOn ` C ) )
48	1, 47	sylib 208	. . . . . . . . . . 11 |- ( ph -> J e. (TopOn ` C ) )
49	48	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. C ) -> J e. (TopOn ` C ) )
50	4	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. C ) -> A C_ C )
51		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. C ) -> x e. C )
52		trnei 21696	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` C ) /\ A C_ C /\ x e. C ) -> ( x e. ( ( cls ` J ) ` A ) <-> ( ( ( nei ` J ) ` { x } ) ↾t A ) e. ( Fil ` A ) ) )
53	52	biimpa 501	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` C ) /\ A C_ C /\ x e. C ) /\ x e. ( ( cls ` J ) ` A ) ) -> ( ( ( nei ` J ) ` { x } ) ↾t A ) e. ( Fil ` A ) )
54	49, 50, 51, 12, 53	syl31anc 1329	. . . . . . . . 9 |- ( ( ph /\ x e. C ) -> ( ( ( nei ` J ) ` { x } ) ↾t A ) e. ( Fil ` A ) )
55	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. C ) -> F : A --> B )
56		flfelbas 21798	. . . . . . . . . . 11 |- ( ( ( K e. (TopOn ` B ) /\ ( ( ( nei ` J ) ` { x } ) ↾t A ) e. ( Fil ` A ) /\ F : A --> B ) /\ y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) -> y e. B )
57	56	ex 450	. . . . . . . . . 10 |- ( ( K e. (TopOn ` B ) /\ ( ( ( nei ` J ) ` { x } ) ↾t A ) e. ( Fil ` A ) /\ F : A --> B ) -> ( y e. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) -> y e. B ) )
58	57	ssrdv 3609	. . . . . . . . 9 |- ( ( K e. (TopOn ` B ) /\ ( ( ( nei ` J ) ` { x } ) ↾t A ) e. ( Fil ` A ) /\ F : A --> B ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) C_ B )
59	46, 54, 55, 58	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ x e. C ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) C_ B )
60	43, 59	syldan 487	. . . . . . 7 |- ( ( ph /\ x e. ( ( cls ` J ) ` A ) ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) C_ B )
61	42, 60	syl5eqss 3649	. . . . . 6 |- ( ( ph /\ x e. ( ( cls ` J ) ` A ) ) -> ran ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) C_ B )
62	61	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. ( ( cls ` J ) ` A ) ran ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) C_ B )
63		iunss 4561	. . . . 5 |- ( U_ x e. ( ( cls ` J ) ` A ) ran ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) C_ B <-> A. x e. ( ( cls ` J ) ` A ) ran ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) C_ B )
64	62, 63	sylibr 224	. . . 4 |- ( ph -> U_ x e. ( ( cls ` J ) ` A ) ran ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) C_ B )
65	38, 64	syl5eqss 3649	. . 3 |- ( ph -> ran U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) C_ B )
66	37, 65	eqsstrd 3639	. 2 |- ( ph -> ran ( ( JCnExt K ) ` F ) C_ B )
67		df-f 5892	. 2 |- ( ( ( JCnExt K ) ` F ) : C --> B <-> ( ( ( JCnExt K ) ` F ) Fn C /\ ran ( ( JCnExt K ) ` F ) C_ B ) )
68	36, 66, 67	sylanbrc 698	1 |- ( ph -> ( ( JCnExt K ) ` F ) : C --> B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   X. cxp 5112   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   clsccl 20822   neicnei 20901   Hauscha 21112   Filcfil 21649   fLimf cflf 21739  CnExtccnext 21863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-pm 7860  df-rest 16083  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-cnext 21864

This theorem is referenced by:  cnextcn  21871  cnextfres1  21872