Theorem cnextfres 21873

Description: F and its extension by continuity agree on the domain of F. (Contributed by Thierry Arnoux, 29-Aug-2020.)

Hypotheses

Ref	Expression
cnextfres.c	|- C = U. J
cnextfres.b	|- B = U. K
cnextfres.j	|- ( ph -> J e. Top )
cnextfres.k	|- ( ph -> K e. Haus )
cnextfres.a	|- ( ph -> A C_ C )
cnextfres.1	|- ( ph -> F e. ( ( J ↾t A ) Cn K ) )
cnextfres.x	|- ( ph -> X e. A )

Assertion

Ref	Expression
cnextfres	|- ( ph -> ( ( ( JCnExt K ) ` F ) ` X ) = ( F ` X ) )

Proof of Theorem cnextfres

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnextfres.j	. . 3 |- ( ph -> J e. Top )
2		cnextfres.k	. . 3 |- ( ph -> K e. Haus )
3		cnextfres.1	. . . . 5 |- ( ph -> F e. ( ( J ↾t A ) Cn K ) )
4		eqid 2622	. . . . . 6 |- U. ( J ↾t A ) = U. ( J ↾t A )
5		cnextfres.b	. . . . . 6 |- B = U. K
6	4, 5	cnf 21050	. . . . 5 |- ( F e. ( ( J ↾t A ) Cn K ) -> F : U. ( J ↾t A ) --> B )
7	3, 6	syl 17	. . . 4 |- ( ph -> F : U. ( J ↾t A ) --> B )
8		cnextfres.a	. . . . . 6 |- ( ph -> A C_ C )
9		cnextfres.c	. . . . . . 7 |- C = U. J
10	9	restuni 20966	. . . . . 6 |- ( ( J e. Top /\ A C_ C ) -> A = U. ( J ↾t A ) )
11	1, 8, 10	syl2anc 693	. . . . 5 |- ( ph -> A = U. ( J ↾t A ) )
12	11	feq2d 6031	. . . 4 |- ( ph -> ( F : A --> B <-> F : U. ( J ↾t A ) --> B ) )
13	7, 12	mpbird 247	. . 3 |- ( ph -> F : A --> B )
14	9, 5	cnextfun 21868	. . 3 |- ( ( ( J e. Top /\ K e. Haus ) /\ ( F : A --> B /\ A C_ C ) ) -> Fun ( ( JCnExt K ) ` F ) )
15	1, 2, 13, 8, 14	syl22anc 1327	. 2 |- ( ph -> Fun ( ( JCnExt K ) ` F ) )
16	9	sscls 20860	. . . . . . . 8 |- ( ( J e. Top /\ A C_ C ) -> A C_ ( ( cls ` J ) ` A ) )
17	1, 8, 16	syl2anc 693	. . . . . . 7 |- ( ph -> A C_ ( ( cls ` J ) ` A ) )
18		cnextfres.x	. . . . . . 7 |- ( ph -> X e. A )
19	17, 18	sseldd 3604	. . . . . 6 |- ( ph -> X e. ( ( cls ` J ) ` A ) )
20	9, 5, 1, 8, 3, 18	flfcntr 21847	. . . . . 6 |- ( ph -> ( F ` X ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) )
21	19, 20	jca 554	. . . . 5 |- ( ph -> ( X e. ( ( cls ` J ) ` A ) /\ ( F ` X ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) ) )
22		sneq 4187	. . . . . . . . . 10 |- ( x = X -> { x } = { X } )
23	22	fveq2d 6195	. . . . . . . . 9 |- ( x = X -> ( ( nei ` J ) ` { x } ) = ( ( nei ` J ) ` { X } ) )
24	23	oveq1d 6665	. . . . . . . 8 |- ( x = X -> ( ( ( nei ` J ) ` { x } ) ↾t A ) = ( ( ( nei ` J ) ` { X } ) ↾t A ) )
25	24	oveq2d 6666	. . . . . . 7 |- ( x = X -> ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) = ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
26	25	fveq1d 6193	. . . . . 6 |- ( x = X -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) = ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) )
27	26	opeliunxp2 5260	. . . . 5 |- ( <. X , ( F ` X ) >. e. U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) <-> ( X e. ( ( cls ` J ) ` A ) /\ ( F ` X ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) ) )
28	21, 27	sylibr 224	. . . 4 |- ( ph -> <. X , ( F ` X ) >. e. U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
29		haustop 21135	. . . . . . 7 |- ( K e. Haus -> K e. Top )
30	2, 29	syl 17	. . . . . 6 |- ( ph -> K e. Top )
31	9, 5	cnextfval 21866	. . . . . 6 |- ( ( ( 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 ) ) )
32	1, 30, 13, 8, 31	syl22anc 1327	. . . . 5 |- ( ph -> ( ( JCnExt K ) ` F ) = U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
33	32	eleq2d 2687	. . . 4 |- ( ph -> ( <. X , ( F ` X ) >. e. ( ( JCnExt K ) ` F ) <-> <. X , ( F ` X ) >. e. U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) ) )
34	28, 33	mpbird 247	. . 3 |- ( ph -> <. X , ( F ` X ) >. e. ( ( JCnExt K ) ` F ) )
35		df-br 4654	. . 3 |- ( X ( ( JCnExt K ) ` F ) ( F ` X ) <-> <. X , ( F ` X ) >. e. ( ( JCnExt K ) ` F ) )
36	34, 35	sylibr 224	. 2 |- ( ph -> X ( ( JCnExt K ) ` F ) ( F ` X ) )
37		funbrfv 6234	. . 3 |- ( Fun ( ( JCnExt K ) ` F ) -> ( X ( ( JCnExt K ) ` F ) ( F ` X ) -> ( ( ( JCnExt K ) ` F ) ` X ) = ( F ` X ) ) )
38	37	imp 445	. 2 |- ( ( Fun ( ( JCnExt K ) ` F ) /\ X ( ( JCnExt K ) ` F ) ( F ` X ) ) -> ( ( ( JCnExt K ) ` F ) ` X ) = ( F ` X ) )
39	15, 36, 38	syl2anc 693	1 |- ( ph -> ( ( ( JCnExt K ) ` F ) ` X ) = ( F ` X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   X. cxp 5112   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698   clsccl 20822   neicnei 20901   Cn ccn 21028   Hauscha 21112   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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  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-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-cnext 21864

This theorem is referenced by:  rrhqima  30058