Theorem cnextfval 21125

Description: The continuous extension of a given function F. (Contributed by Thierry Arnoux, 1-Dec-2017.)

Hypotheses

Ref	Expression
cnextfval.1	|- X = U. J
cnextfval.2	|- B = U. K

Assertion

Ref	Expression
cnextfval	|- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> ( ( JCnExt K ) ` F ) = U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )

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

Proof of Theorem cnextfval

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnextval 21124	. . 3 |- ( ( J e. Top /\ K e. Top ) -> ( JCnExt K ) = ( f e. ( U. K ^pm U. J ) |-> U_ x e. ( ( cls ` J ) ` dom f ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) ) )
2	1	adantr 471	. 2 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> ( JCnExt K ) = ( f e. ( U. K ^pm U. J ) |-> U_ x e. ( ( cls ` J ) ` dom f ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) ) )
3		simpr 467	. . . . . 6 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> f = F )
4	3	dmeqd 5055	. . . . 5 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> dom f = dom F )
5		simplrl 775	. . . . . 6 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> F : A --> B )
6		fdm 5755	. . . . . 6 |- ( F : A --> B -> dom F = A )
7	5, 6	syl 17	. . . . 5 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> dom F = A )
8	4, 7	eqtrd 2495	. . . 4 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> dom f = A )
9	8	fveq2d 5891	. . 3 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> ( ( cls ` J ) ` dom f ) = ( ( cls ` J ) ` A ) )
10	8	oveq2d 6330	. . . . . 6 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> ( ( ( nei ` J ) ` { x } ) ↾t dom f ) = ( ( ( nei ` J ) ` { x } ) ↾t A ) )
11	10	oveq2d 6330	. . . . 5 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) = ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) )
12	11, 3	fveq12d 5893	. . . 4 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) = ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) )
13	12	xpeq2d 4876	. . 3 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) = ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
14	9, 13	iuneq12d 4317	. 2 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> U_ x e. ( ( cls ` J ) ` dom f ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) = U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
15		uniexg 6614	. . . 4 |- ( K e. Top -> U. K e. _V )
16	15	ad2antlr 738	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> U. K e. _V )
17		uniexg 6614	. . . 4 |- ( J e. Top -> U. J e. _V )
18	17	ad2antrr 737	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> U. J e. _V )
19		eqid 2461	. . . . . 6 |- A = A
20		cnextfval.2	. . . . . 6 |- B = U. K
21	19, 20	feq23i 5744	. . . . 5 |- ( F : A --> B <-> F : A --> U. K )
22	21	biimpi 199	. . . 4 |- ( F : A --> B -> F : A --> U. K )
23	22	ad2antrl 739	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> F : A --> U. K )
24		cnextfval.1	. . . . . 6 |- X = U. J
25	24	sseq2i 3468	. . . . 5 |- ( A C_ X <-> A C_ U. J )
26	25	biimpi 199	. . . 4 |- ( A C_ X -> A C_ U. J )
27	26	ad2antll 740	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> A C_ U. J )
28		elpm2r 7514	. . 3 |- ( ( ( U. K e. _V /\ U. J e. _V ) /\ ( F : A --> U. K /\ A C_ U. J ) ) -> F e. ( U. K ^pm U. J ) )
29	16, 18, 23, 27, 28	syl22anc 1277	. 2 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> F e. ( U. K ^pm U. J ) )
30		fvex 5897	. . . 4 |- ( ( cls ` J ) ` A ) e. _V
31		snex 4654	. . . . 5 |- { x } e. _V
32		fvex 5897	. . . . 5 |- ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) e. _V
33	31, 32	xpex 6621	. . . 4 |- ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) e. _V
34	30, 33	iunex 6799	. . 3 |- U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) e. _V
35	34	a1i 11	. 2 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) e. _V )
36	2, 14, 29, 35	fvmptd 5976	1 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> ( ( JCnExt K ) ` F ) = U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1454   e. wcel 1897   _Vcvv 3056   C_ wss 3415   {csn 3979   U.cuni 4211   U_ciun 4291   |-> cmpt 4474   X. cxp 4850   dom cdm 4852   -->wf 5596   ` cfv 5600  (class class class)co 6314   ^pm cpm 7498   ↾_t crest 15367   Topctop 19965   clsccl 20081   neicnei 20161   fLimf cflf 20998  CnExtccnext 21122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-pm 7500  df-cnext 21123

This theorem is referenced by:  cnextrel  21126  cnextfun  21127  cnextfvval  21128  cnextf  21129  cnextfres  21132