Theorem cnextfval 20428

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 20427	. . 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 465	. 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 461	. . . . . 6 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> f = F )
4	3	dmeqd 5211	. . . . 5 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> dom f = dom F )
5		simplrl 759	. . . . . 6 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> F : A --> B )
6		fdm 5741	. . . . . 6 |- ( F : A --> B -> dom F = A )
7	5, 6	syl 16	. . . . 5 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> dom F = A )
8	4, 7	eqtrd 2508	. . . 4 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> dom f = A )
9	8	fveq2d 5876	. . 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 6311	. . . . . 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 6311	. . . . 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 5878	. . . 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 5029	. . 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 4357	. 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 6592	. . . 4 |- ( K e. Top -> U. K e. _V )
16	15	ad2antlr 726	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> U. K e. _V )
17		uniexg 6592	. . . 4 |- ( J e. Top -> U. J e. _V )
18	17	ad2antrr 725	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> U. J e. _V )
19		eqid 2467	. . . . . 6 |- A = A
20		cnextfval.2	. . . . . 6 |- B = U. K
21	19, 20	feq23i 5731	. . . . 5 |- ( F : A --> B <-> F : A --> U. K )
22	21	biimpi 194	. . . 4 |- ( F : A --> B -> F : A --> U. K )
23	22	ad2antrl 727	. . 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 3534	. . . . 5 |- ( A C_ X <-> A C_ U. J )
26	25	biimpi 194	. . . 4 |- ( A C_ X -> A C_ U. J )
27	26	ad2antll 728	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> A C_ U. J )
28		elpm2r 7448	. . 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 1229	. 2 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> F e. ( U. K ^pm U. J ) )
30		fvex 5882	. . . 4 |- ( ( cls ` J ) ` A ) e. _V
31		snex 4694	. . . . 5 |- { x } e. _V
32		fvex 5882	. . . . 5 |- ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) e. _V
33	31, 32	xpex 6599	. . . 4 |- ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) e. _V
34	30, 33	iunex 6775	. . 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 5962	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 369   = wceq 1379   e. wcel 1767   _Vcvv 3118   C_ wss 3481   {csn 4033   U.cuni 4251   U_ciun 4331   |-> cmpt 4511   X. cxp 5003   dom cdm 5005   -->wf 5590   ` cfv 5594  (class class class)co 6295   ^pm cpm 7433   ↾_t crest 14692   Topctop 19261   clsccl 19385   neicnei 19464   fLimf cflf 20302  CnExtccnext 20425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-pm 7435  df-cnext 20426

This theorem is referenced by:  cnextrel  20429  cnextfun  20430  cnextfvval  20431  cnextf  20432