Theorem xkoval 21390

Description: Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypotheses

Ref	Expression
xkoval.x	|- X = U. R
xkoval.k	|- K = { x e. ~P X | ( R ↾t x ) e. Comp }
xkoval.t	|- T = ( k e. K , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } )

Assertion

Ref	Expression
xkoval	|- ( ( R e. Top /\ S e. Top ) -> ( S ^ko R ) = ( topGen ` ( fi ` ran T ) ) )

Distinct variable groups:   v, k, K   f, k, v, x, R   S, f, k, v, x   k, X, x

Allowed substitution hints:   T( x, v, f, k)   K( x, f)   X( v, f)

Proof of Theorem xkoval

Dummy variables s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . . . . . . 13 |- ( ( s = S /\ r = R ) -> r = R )
2	1	unieqd 4446	. . . . . . . . . . . 12 |- ( ( s = S /\ r = R ) -> U. r = U. R )
3		xkoval.x	. . . . . . . . . . . 12 |- X = U. R
4	2, 3	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( s = S /\ r = R ) -> U. r = X )
5	4	pweqd 4163	. . . . . . . . . 10 |- ( ( s = S /\ r = R ) -> ~P U. r = ~P X )
6	1	oveq1d 6665	. . . . . . . . . . 11 |- ( ( s = S /\ r = R ) -> ( r ↾t x ) = ( R ↾t x ) )
7	6	eleq1d 2686	. . . . . . . . . 10 |- ( ( s = S /\ r = R ) -> ( ( r ↾t x ) e. Comp <-> ( R ↾t x ) e. Comp ) )
8	5, 7	rabeqbidv 3195	. . . . . . . . 9 |- ( ( s = S /\ r = R ) -> { x e. ~P U. r | ( r ↾t x ) e. Comp } = { x e. ~P X | ( R ↾t x ) e. Comp } )
9		xkoval.k	. . . . . . . . 9 |- K = { x e. ~P X | ( R ↾t x ) e. Comp }
10	8, 9	syl6eqr 2674	. . . . . . . 8 |- ( ( s = S /\ r = R ) -> { x e. ~P U. r | ( r ↾t x ) e. Comp } = K )
11		simpl 473	. . . . . . . 8 |- ( ( s = S /\ r = R ) -> s = S )
12	1, 11	oveq12d 6668	. . . . . . . . 9 |- ( ( s = S /\ r = R ) -> ( r Cn s ) = ( R Cn S ) )
13		rabeq 3192	. . . . . . . . 9 |- ( ( r Cn s ) = ( R Cn S ) -> { f e. ( r Cn s ) | ( f " k ) C_ v } = { f e. ( R Cn S ) | ( f " k ) C_ v } )
14	12, 13	syl 17	. . . . . . . 8 |- ( ( s = S /\ r = R ) -> { f e. ( r Cn s ) | ( f " k ) C_ v } = { f e. ( R Cn S ) | ( f " k ) C_ v } )
15	10, 11, 14	mpt2eq123dv 6717	. . . . . . 7 |- ( ( s = S /\ r = R ) -> ( k e. { x e. ~P U. r | ( r ↾t x ) e. Comp } , v e. s |-> { f e. ( r Cn s ) | ( f " k ) C_ v } ) = ( k e. K , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } ) )
16		xkoval.t	. . . . . . 7 |- T = ( k e. K , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } )
17	15, 16	syl6eqr 2674	. . . . . 6 |- ( ( s = S /\ r = R ) -> ( k e. { x e. ~P U. r | ( r ↾t x ) e. Comp } , v e. s |-> { f e. ( r Cn s ) | ( f " k ) C_ v } ) = T )
18	17	rneqd 5353	. . . . 5 |- ( ( s = S /\ r = R ) -> ran ( k e. { x e. ~P U. r | ( r ↾t x ) e. Comp } , v e. s |-> { f e. ( r Cn s ) | ( f " k ) C_ v } ) = ran T )
19	18	fveq2d 6195	. . . 4 |- ( ( s = S /\ r = R ) -> ( fi ` ran ( k e. { x e. ~P U. r | ( r ↾t x ) e. Comp } , v e. s |-> { f e. ( r Cn s ) | ( f " k ) C_ v } ) ) = ( fi ` ran T ) )
20	19	fveq2d 6195	. . 3 |- ( ( s = S /\ r = R ) -> ( topGen ` ( fi ` ran ( k e. { x e. ~P U. r | ( r ↾t x ) e. Comp } , v e. s |-> { f e. ( r Cn s ) | ( f " k ) C_ v } ) ) ) = ( topGen ` ( fi ` ran T ) ) )
21		df-xko 21366	. . 3 |- ^ko = ( s e. Top , r e. Top |-> ( topGen ` ( fi ` ran ( k e. { x e. ~P U. r | ( r ↾t x ) e. Comp } , v e. s |-> { f e. ( r Cn s ) | ( f " k ) C_ v } ) ) ) )
22		fvex 6201	. . 3 |- ( topGen ` ( fi ` ran T ) ) e. _V
23	20, 21, 22	ovmpt2a 6791	. 2 |- ( ( S e. Top /\ R e. Top ) -> ( S ^ko R ) = ( topGen ` ( fi ` ran T ) ) )
24	23	ancoms 469	1 |- ( ( R e. Top /\ S e. Top ) -> ( S ^ko R ) = ( topGen ` ( fi ` ran T ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ficfi 8316   ↾_t crest 16081   topGenctg 16098   Topctop 20698   Cn ccn 21028   Compccmp 21189   ^ko cxko 21364

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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-xko 21366

This theorem is referenced by:  xkotop  21391  xkoopn  21392  xkouni  21402  xkoccn  21422  xkopt  21458  xkoco1cn  21460  xkoco2cn  21461  xkococn  21463  xkoinjcn  21490