Theorem cnextfval 21866

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 21865	. . 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 481	. 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 477	. . . . . 6 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> f = F )
4	3	dmeqd 5326	. . . . 5 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> dom f = dom F )
5		simplrl 800	. . . . . 6 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> F : A --> B )
6		fdm 6051	. . . . . 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 2656	. . . 4 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> dom f = A )
9	8	fveq2d 6195	. . 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 6666	. . . . . 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 6666	. . . . 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 6197	. . . 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 5139	. . 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 4546	. 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 6955	. . . 4 |- ( K e. Top -> U. K e. _V )
16	15	ad2antlr 763	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> U. K e. _V )
17		uniexg 6955	. . . 4 |- ( J e. Top -> U. J e. _V )
18	17	ad2antrr 762	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> U. J e. _V )
19		eqid 2622	. . . . . 6 |- A = A
20		cnextfval.2	. . . . . 6 |- B = U. K
21	19, 20	feq23i 6039	. . . . 5 |- ( F : A --> B <-> F : A --> U. K )
22	21	biimpi 206	. . . 4 |- ( F : A --> B -> F : A --> U. K )
23	22	ad2antrl 764	. . 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 3630	. . . . 5 |- ( A C_ X <-> A C_ U. J )
26	25	biimpi 206	. . . 4 |- ( A C_ X -> A C_ U. J )
27	26	ad2antll 765	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> A C_ U. J )
28		elpm2r 7875	. . 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 1327	. 2 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> F e. ( U. K ^pm U. J ) )
30		fvex 6201	. . . 4 |- ( ( cls ` J ) ` A ) e. _V
31		snex 4908	. . . . 5 |- { x } e. _V
32		fvex 6201	. . . . 5 |- ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) e. _V
33	31, 32	xpex 6962	. . . 4 |- ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) e. _V
34	30, 33	iunex 7147	. . 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 6288	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 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   ↾_t crest 16081   Topctop 20698   clsccl 20822   neicnei 20901   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-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-iun 4522  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-pm 7860  df-cnext 21864

This theorem is referenced by:  cnextrel  21867  cnextfun  21868  cnextfvval  21869  cnextf  21870  cnextfres  21873