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Theorem dibintclN 34729
Description: The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibintcl.h  |-  H  =  ( LHyp `  K
)
dibintcl.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibintclN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| S  e.  ran  I )

Proof of Theorem dibintclN
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dibintcl.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
2 dibintcl.i . . . . . . . 8  |-  I  =  ( ( DIsoB `  K
) `  W )
31, 2dibf11N 34723 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
43adantr 467 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I : dom  I -1-1-onto-> ran  I )
5 f1ofn 5813 . . . . . 6  |-  ( I : dom  I -1-1-onto-> ran  I  ->  I  Fn  dom  I
)
64, 5syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I  Fn  dom  I )
7 cnvimass 5187 . . . . 5  |-  ( `' I " S ) 
C_  dom  I
8 fnssres 5687 . . . . 5  |-  ( ( I  Fn  dom  I  /\  ( `' I " S )  C_  dom  I )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
96, 7, 8sylancl 667 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
10 fniinfv 5922 . . . 4  |-  ( ( I  |`  ( `' I " S ) )  Fn  ( `' I " S )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^| ran  ( I  |`  ( `' I " S ) ) )
119, 10syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^| ran  ( I  |`  ( `' I " S ) ) )
12 df-ima 4846 . . . . 5  |-  ( I
" ( `' I " S ) )  =  ran  ( I  |`  ( `' I " S ) )
13 f1ofo 5819 . . . . . . . 8  |-  ( I : dom  I -1-1-onto-> ran  I  ->  I : dom  I -onto-> ran  I )
143, 13syl 17 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -onto-> ran  I )
1514adantr 467 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I : dom  I -onto-> ran  I
)
16 simprl 763 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  C_ 
ran  I )
17 foimacnv 5829 . . . . . 6  |-  ( ( I : dom  I -onto-> ran  I  /\  S  C_  ran  I )  ->  (
I " ( `' I " S ) )  =  S )
1815, 16, 17syl2anc 666 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I " ( `' I " S ) )  =  S )
1912, 18syl5eqr 2498 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ran  ( I  |`  ( `' I " S ) )  =  S )
2019inteqd 4238 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| ran  ( I  |`  ( `' I " S ) )  =  |^| S
)
2111, 20eqtrd 2484 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^| S )
22 simpl 459 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
237a1i 11 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  dom  I )
24 simprr 765 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  =/=  (/) )
25 n0 3740 . . . . . . 7  |-  ( S  =/=  (/)  <->  E. y  y  e.  S )
2624, 25sylib 200 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  E. y 
y  e.  S )
2716sselda 3431 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  y  e.  ran  I
)
283ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  I : dom  I -1-1-onto-> ran  I )
2928, 5syl 17 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  I  Fn  dom  I
)
30 fvelrnb 5910 . . . . . . . . 9  |-  ( I  Fn  dom  I  -> 
( y  e.  ran  I 
<->  E. x  e.  dom  I ( I `  x )  =  y ) )
3129, 30syl 17 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  ( y  e.  ran  I 
<->  E. x  e.  dom  I ( I `  x )  =  y ) )
3227, 31mpbid 214 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  E. x  e.  dom  I ( I `  x )  =  y )
33 f1ofun 5814 . . . . . . . . . . . . . . . 16  |-  ( I : dom  I -1-1-onto-> ran  I  ->  Fun  I )
343, 33syl 17 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
3534adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  Fun  I )
36 fvimacnv 5995 . . . . . . . . . . . . . 14  |-  ( ( Fun  I  /\  x  e.  dom  I )  -> 
( ( I `  x )  e.  S  <->  x  e.  ( `' I " S ) ) )
3735, 36sylan 474 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  x  e.  dom  I )  ->  ( ( I `
 x )  e.  S  <->  x  e.  ( `' I " S ) ) )
38 ne0i 3736 . . . . . . . . . . . . 13  |-  ( x  e.  ( `' I " S )  ->  ( `' I " S )  =/=  (/) )
3937, 38syl6bi 232 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  x  e.  dom  I )  ->  ( ( I `
 x )  e.  S  ->  ( `' I " S )  =/=  (/) ) )
4039ex 436 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
x  e.  dom  I  ->  ( ( I `  x )  e.  S  ->  ( `' I " S )  =/=  (/) ) ) )
41 eleq1 2516 . . . . . . . . . . . . 13  |-  ( ( I `  x )  =  y  ->  (
( I `  x
)  e.  S  <->  y  e.  S ) )
4241biimprd 227 . . . . . . . . . . . 12  |-  ( ( I `  x )  =  y  ->  (
y  e.  S  -> 
( I `  x
)  e.  S ) )
4342imim1d 78 . . . . . . . . . . 11  |-  ( ( I `  x )  =  y  ->  (
( ( I `  x )  e.  S  ->  ( `' I " S )  =/=  (/) )  -> 
( y  e.  S  ->  ( `' I " S )  =/=  (/) ) ) )
4440, 43syl9 73 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( I `  x
)  =  y  -> 
( x  e.  dom  I  ->  ( y  e.  S  ->  ( `' I " S )  =/=  (/) ) ) ) )
4544com24 90 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
y  e.  S  -> 
( x  e.  dom  I  ->  ( ( I `
 x )  =  y  ->  ( `' I " S )  =/=  (/) ) ) ) )
4645imp 431 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  ( x  e.  dom  I  ->  ( ( I `
 x )  =  y  ->  ( `' I " S )  =/=  (/) ) ) )
4746rexlimdv 2876 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  ( E. x  e. 
dom  I ( I `
 x )  =  y  ->  ( `' I " S )  =/=  (/) ) )
4832, 47mpd 15 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  ( `' I " S )  =/=  (/) )
4926, 48exlimddv 1780 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S )  =/=  (/) )
50 eqid 2450 . . . . . 6  |-  ( glb `  K )  =  ( glb `  K )
5150, 1, 2dibglbN 34728 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( `' I " S ) 
C_  dom  I  /\  ( `' I " S )  =/=  (/) ) )  -> 
( I `  (
( glb `  K
) `  ( `' I " S ) ) )  =  |^|_ y  e.  ( `' I " S ) ( I `
 y ) )
5222, 23, 49, 51syl12anc 1265 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I `  ( ( glb `  K ) `  ( `' I " S ) ) )  =  |^|_ y  e.  ( `' I " S ) ( I `  y ) )
53 fvres 5877 . . . . 5  |-  ( y  e.  ( `' I " S )  ->  (
( I  |`  ( `' I " S ) ) `  y )  =  ( I `  y ) )
5453iineq2i 4297 . . . 4  |-  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^|_ y  e.  ( `' I " S ) ( I `  y
)
5552, 54syl6eqr 2502 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I `  ( ( glb `  K ) `  ( `' I " S ) ) )  =  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `  y ) )
56 hlclat 32918 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CLat )
5756ad2antrr 731 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  K  e.  CLat )
58 eqid 2450 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
59 eqid 2450 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
6058, 59, 1, 2dibdmN 34719 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  =  {
x  e.  ( Base `  K )  |  x ( le `  K
) W } )
61 ssrab2 3513 . . . . . . . . 9  |-  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  C_  ( Base `  K )
6260, 61syl6eqss 3481 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  C_  ( Base `  K ) )
6362adantr 467 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  dom  I  C_  ( Base `  K
) )
647, 63syl5ss 3442 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  ( Base `  K
) )
6558, 50clatglbcl 16353 . . . . . 6  |-  ( ( K  e.  CLat  /\  ( `' I " S ) 
C_  ( Base `  K
) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
6657, 64, 65syl2anc 666 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
67 n0 3740 . . . . . . 7  |-  ( ( `' I " S )  =/=  (/)  <->  E. y  y  e.  ( `' I " S ) )
6849, 67sylib 200 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  E. y 
y  e.  ( `' I " S ) )
69 hllat 32923 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
7069ad3antrrr 735 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  K  e.  Lat )
7166adantr 467 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) )  e.  ( Base `  K ) )
7264sselda 3431 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  (
Base `  K )
)
7358, 1lhpbase 33557 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7473ad3antlr 736 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  W  e.  (
Base `  K )
)
7556ad3antrrr 735 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  K  e.  CLat )
7660adantr 467 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  dom  I  =  { x  e.  ( Base `  K
)  |  x ( le `  K ) W } )
777, 76syl5sseq 3479 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  { x  e.  ( Base `  K
)  |  x ( le `  K ) W } )
7877, 61syl6ss 3443 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  ( Base `  K
) )
7978adantr 467 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( `' I " S )  C_  ( Base `  K ) )
80 simpr 463 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  ( `' I " S ) )
8158, 59, 50clatglble 16364 . . . . . . . 8  |-  ( ( K  e.  CLat  /\  ( `' I " S ) 
C_  ( Base `  K
)  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) y )
8275, 79, 80, 81syl3anc 1267 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) y )
837sseli 3427 . . . . . . . . . 10  |-  ( y  e.  ( `' I " S )  ->  y  e.  dom  I )
8483adantl 468 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  dom  I )
8558, 59, 1, 2dibeldmN 34720 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( y  e.  dom  I 
<->  ( y  e.  (
Base `  K )  /\  y ( le `  K ) W ) ) )
8685ad2antrr 731 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( y  e. 
dom  I  <->  ( y  e.  ( Base `  K
)  /\  y ( le `  K ) W ) ) )
8784, 86mpbid 214 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( y  e.  ( Base `  K
)  /\  y ( le `  K ) W ) )
8887simprd 465 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y ( le
`  K ) W )
8958, 59, 70, 71, 72, 74, 82, 88lattrd 16297 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) W )
9068, 89exlimddv 1780 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) ) ( le `  K
) W )
9158, 59, 1, 2dibeldmN 34720 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( glb `  K ) `  ( `' I " S ) )  e.  dom  I  <->  ( ( ( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
)  /\  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) W ) ) )
9291adantr 467 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( ( glb `  K
) `  ( `' I " S ) )  e.  dom  I  <->  ( (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
)  /\  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) W ) ) )
9366, 90, 92mpbir2and 932 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  dom  I )
941, 2dibclN 34724 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( glb `  K ) `  ( `' I " S ) )  e.  dom  I
)  ->  ( I `  ( ( glb `  K
) `  ( `' I " S ) ) )  e.  ran  I
)
9593, 94syldan 473 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I `  ( ( glb `  K ) `  ( `' I " S ) ) )  e.  ran  I )
9655, 95eqeltrrd 2529 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  e. 
ran  I )
9721, 96eqeltrrd 2529 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| S  e.  ran  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443   E.wex 1662    e. wcel 1886    =/= wne 2621   E.wrex 2737   {crab 2740    C_ wss 3403   (/)c0 3730   |^|cint 4233   |^|_ciin 4278   class class class wbr 4401   `'ccnv 4832   dom cdm 4833   ran crn 4834    |` cres 4835   "cima 4836   Fun wfun 5575    Fn wfn 5576   -onto->wfo 5579   -1-1-onto->wf1o 5580   ` cfv 5581   Basecbs 15114   lecple 15190   glbcglb 16181   Latclat 16284   CLatccla 16346   HLchlt 32910   LHypclh 33543   DIsoBcdib 34700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-riotaBAD 32519
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-undef 7017  df-map 7471  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-oposet 32736  df-ol 32738  df-oml 32739  df-covers 32826  df-ats 32827  df-atl 32858  df-cvlat 32882  df-hlat 32911  df-llines 33057  df-lplanes 33058  df-lvols 33059  df-lines 33060  df-psubsp 33062  df-pmap 33063  df-padd 33355  df-lhyp 33547  df-laut 33548  df-ldil 33663  df-ltrn 33664  df-trl 33719  df-disoa 34591  df-dib 34701
This theorem is referenced by: (None)
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