Theorem cnextval 21865

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

Assertion

Ref	Expression
cnextval	|- ( ( 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 ) ) ) )

Distinct variable groups:   x, f, J   f, K, x

Proof of Theorem cnextval

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4444	. . . 4 |- ( j = J -> U. j = U. J )
2	1	oveq2d 6666	. . 3 |- ( j = J -> ( U. k ^pm U. j ) = ( U. k ^pm U. J ) )
3		fveq2 6191	. . . . 5 |- ( j = J -> ( cls ` j ) = ( cls ` J ) )
4	3	fveq1d 6193	. . . 4 |- ( j = J -> ( ( cls ` j ) ` dom f ) = ( ( cls ` J ) ` dom f ) )
5		fveq2 6191	. . . . . . . . 9 |- ( j = J -> ( nei ` j ) = ( nei ` J ) )
6	5	fveq1d 6193	. . . . . . . 8 |- ( j = J -> ( ( nei ` j ) ` { x } ) = ( ( nei ` J ) ` { x } ) )
7	6	oveq1d 6665	. . . . . . 7 |- ( j = J -> ( ( ( nei ` j ) ` { x } ) ↾t dom f ) = ( ( ( nei ` J ) ` { x } ) ↾t dom f ) )
8	7	oveq2d 6666	. . . . . 6 |- ( j = J -> ( k fLimf ( ( ( nei ` j ) ` { x } ) ↾t dom f ) ) = ( k fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) )
9	8	fveq1d 6193	. . . . 5 |- ( j = J -> ( ( k fLimf ( ( ( nei ` j ) ` { x } ) ↾t dom f ) ) ` f ) = ( ( k fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) )
10	9	xpeq2d 5139	. . . 4 |- ( j = J -> ( { x } X. ( ( k fLimf ( ( ( nei ` j ) ` { x } ) ↾t dom f ) ) ` f ) ) = ( { x } X. ( ( k fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) )
11	4, 10	iuneq12d 4546	. . 3 |- ( j = J -> U_ x e. ( ( cls ` j ) ` dom f ) ( { x } X. ( ( k fLimf ( ( ( nei ` j ) ` { x } ) ↾t dom f ) ) ` f ) ) = U_ x e. ( ( cls ` J ) ` dom f ) ( { x } X. ( ( k fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) )
12	2, 11	mpteq12dv 4733	. 2 |- ( j = J -> ( 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 ) ) ) = ( 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 ) ) ) )
13		unieq 4444	. . . 4 |- ( k = K -> U. k = U. K )
14	13	oveq1d 6665	. . 3 |- ( k = K -> ( U. k ^pm U. J ) = ( U. K ^pm U. J ) )
15		oveq1 6657	. . . . . 6 |- ( k = K -> ( k fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) = ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) )
16	15	fveq1d 6193	. . . . 5 |- ( k = K -> ( ( k fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) = ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) )
17	16	xpeq2d 5139	. . . 4 |- ( k = K -> ( { x } X. ( ( k fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) = ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) )
18	17	iuneq2d 4547	. . 3 |- ( k = K -> U_ x e. ( ( cls ` J ) ` dom f ) ( { x } X. ( ( k fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) = U_ x e. ( ( cls ` J ) ` dom f ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t dom f ) ) ` f ) ) )
19	14, 18	mpteq12dv 4733	. 2 |- ( k = 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 ) ) ) = ( 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 ) ) ) )
20		df-cnext 21864	. 2 |- CnExt = ( j e. Top , k e. Top |-> ( 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 ) ) ) )
21		ovex 6678	. . 3 |- ( U. K ^pm U. J ) e. _V
22	21	mptex 6486	. 2 |- ( 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 ) ) ) e. _V
23	12, 19, 20, 22	ovmpt2 6796	1 |- ( ( 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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ` 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-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-pr 4906

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-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-cnext 21864

This theorem is referenced by:  cnextfval  21866