Theorem xkobval 21389

Description: Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-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
xkobval	|- ran T = { s | E. k e. ~P X E. v e. S ( ( R ↾t k ) e. Comp /\ s = { f e. ( R Cn S ) | ( f " k ) C_ v } ) }

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

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

Proof of Theorem xkobval

Step	Hyp	Ref	Expression
1		xkoval.t	. . 3 |- T = ( k e. K , v e. S |-> { f e. ( R Cn S ) | ( f " k ) C_ v } )
2	1	rnmpt2 6770	. 2 |- ran T = { s | E. k e. K E. v e. S s = { f e. ( R Cn S ) | ( f " k ) C_ v } }
3		oveq2 6658	. . . . . 6 |- ( x = k -> ( R ↾t x ) = ( R ↾t k ) )
4	3	eleq1d 2686	. . . . 5 |- ( x = k -> ( ( R ↾t x ) e. Comp <-> ( R ↾t k ) e. Comp ) )
5	4	rexrab 3370	. . . 4 |- ( E. k e. { x e. ~P X | ( R ↾t x ) e. Comp } E. v e. S s = { f e. ( R Cn S ) | ( f " k ) C_ v } <-> E. k e. ~P X ( ( R ↾t k ) e. Comp /\ E. v e. S s = { f e. ( R Cn S ) | ( f " k ) C_ v } ) )
6		xkoval.k	. . . . 5 |- K = { x e. ~P X | ( R ↾t x ) e. Comp }
7	6	rexeqi 3143	. . . 4 |- ( E. k e. K E. v e. S s = { f e. ( R Cn S ) | ( f " k ) C_ v } <-> E. k e. { x e. ~P X | ( R ↾t x ) e. Comp } E. v e. S s = { f e. ( R Cn S ) | ( f " k ) C_ v } )
8		r19.42v 3092	. . . . 5 |- ( E. v e. S ( ( R ↾t k ) e. Comp /\ s = { f e. ( R Cn S ) | ( f " k ) C_ v } ) <-> ( ( R ↾t k ) e. Comp /\ E. v e. S s = { f e. ( R Cn S ) | ( f " k ) C_ v } ) )
9	8	rexbii 3041	. . . 4 |- ( E. k e. ~P X E. v e. S ( ( R ↾t k ) e. Comp /\ s = { f e. ( R Cn S ) | ( f " k ) C_ v } ) <-> E. k e. ~P X ( ( R ↾t k ) e. Comp /\ E. v e. S s = { f e. ( R Cn S ) | ( f " k ) C_ v } ) )
10	5, 7, 9	3bitr4i 292	. . 3 |- ( E. k e. K E. v e. S s = { f e. ( R Cn S ) | ( f " k ) C_ v } <-> E. k e. ~P X E. v e. S ( ( R ↾t k ) e. Comp /\ s = { f e. ( R Cn S ) | ( f " k ) C_ v } ) )
11	10	abbii 2739	. 2 |- { s | E. k e. K E. v e. S s = { f e. ( R Cn S ) | ( f " k ) C_ v } } = { s | E. k e. ~P X E. v e. S ( ( R ↾t k ) e. Comp /\ s = { f e. ( R Cn S ) | ( f " k ) C_ v } ) }
12	2, 11	eqtri 2644	1 |- ran T = { s | E. k e. ~P X E. v e. S ( ( R ↾t k ) e. Comp /\ s = { f e. ( R Cn S ) | ( f " k ) C_ v } ) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ran crn 5115   "cima 5117  (class class class)co 6650   |-> cmpt2 6652   ↾_t crest 16081   Cn ccn 21028   Compccmp 21189

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-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-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  xkoccn  21422  xkoco1cn  21460  xkoco2cn  21461  xkoinjcn  21490