Theorem cnextfval 19639

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 19638	. . 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 5047	. . . . 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 5568	. . . . . 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 2475	. . . 4 |- ( ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) /\ f = F ) -> dom f = A )
9	8	fveq2d 5700	. . 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 6112	. . . . . 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 6112	. . . . 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 5702	. . . 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 4869	. . 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 4201	. 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 6382	. . . 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 6382	. . . 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 2443	. . . . . 6 |- A = A
20		cnextfval.2	. . . . . 6 |- B = U. K
21	19, 20	feq23i 5558	. . . . 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 3386	. . . . 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 7235	. . 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 1219	. 2 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ X ) ) -> F e. ( U. K ^pm U. J ) )
30		fvex 5706	. . . 4 |- ( ( cls ` J ) ` A ) e. _V
31		snex 4538	. . . . 5 |- { x } e. _V
32		fvex 5706	. . . . 5 |- ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) e. _V
33	31, 32	xpex 6513	. . . 4 |- ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) e. _V
34	30, 33	iunex 6562	. . 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 5784	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 1369   e. wcel 1756   _Vcvv 2977   C_ wss 3333   {csn 3882   U.cuni 4096   U_ciun 4176   e. cmpt 4355   X. cxp 4843   dom cdm 4845   -->wf 5419   ` cfv 5423  (class class class)co 6096   ^pm cpm 7220   ↾_t crest 14364   Topctop 18503   clsccl 18627   neicnei 18706   fLimf cflf 19513  CnExtccnext 19636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-pm 7222  df-cnext 19637

This theorem is referenced by:  cnextrel  19640  cnextfun  19641  cnextfvval  19642  cnextf  19643