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Theorem dibintclN 32039
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 32033 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
43adantr 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I : dom  I -1-1-onto-> ran  I )
5 f1ofn 5678 . . . . . 6  |-  ( I : dom  I -1-1-onto-> ran  I  ->  I  Fn  dom  I
)
64, 5syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I  Fn  dom  I )
7 cnvimass 5227 . . . . 5  |-  ( `' I " S ) 
C_  dom  I
8 fnssres 5561 . . . . 5  |-  ( ( I  Fn  dom  I  /\  ( `' I " S )  C_  dom  I )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
96, 7, 8sylancl 645 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
10 fniinfv 5788 . . . 4  |-  ( ( I  |`  ( `' I " S ) )  Fn  ( `' I " S )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^| ran  ( I  |`  ( `' I " S ) ) )
119, 10syl 16 . . 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 4894 . . . . 5  |-  ( I
" ( `' I " S ) )  =  ran  ( I  |`  ( `' I " S ) )
13 f1ofo 5684 . . . . . . . 8  |-  ( I : dom  I -1-1-onto-> ran  I  ->  I : dom  I -onto-> ran  I )
143, 13syl 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -onto-> ran  I )
1514adantr 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I : dom  I -onto-> ran  I
)
16 simprl 734 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  C_ 
ran  I )
17 foimacnv 5695 . . . . . 6  |-  ( ( I : dom  I -onto-> ran  I  /\  S  C_  ran  I )  ->  (
I " ( `' I " S ) )  =  S )
1815, 16, 17syl2anc 644 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I " ( `' I " S ) )  =  S )
1912, 18syl5eqr 2484 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ran  ( I  |`  ( `' I " S ) )  =  S )
2019inteqd 4057 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| ran  ( I  |`  ( `' I " S ) )  =  |^| S
)
2111, 20eqtrd 2470 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^| S )
22 simpl 445 . . . . 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 735 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  =/=  (/) )
25 n0 3639 . . . . . . 7  |-  ( S  =/=  (/)  <->  E. y  y  e.  S )
2624, 25sylib 190 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  E. y 
y  e.  S )
2716sselda 3350 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  y  e.  ran  I
)
283ad2antrr 708 . . . . . . . . . 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 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  I  Fn  dom  I
)
30 fvelrnb 5777 . . . . . . . . 9  |-  ( I  Fn  dom  I  -> 
( y  e.  ran  I 
<->  E. x  e.  dom  I ( I `  x )  =  y ) )
3129, 30syl 16 . . . . . . . 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 203 . . . . . . 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 5679 . . . . . . . . . . . . . . . 16  |-  ( I : dom  I -1-1-onto-> ran  I  ->  Fun  I )
343, 33syl 16 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
3534adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  Fun  I )
36 fvimacnv 5848 . . . . . . . . . . . . . 14  |-  ( ( Fun  I  /\  x  e.  dom  I )  -> 
( ( I `  x )  e.  S  <->  x  e.  ( `' I " S ) ) )
3735, 36sylan 459 . . . . . . . . . . . . 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 3636 . . . . . . . . . . . . 13  |-  ( x  e.  ( `' I " S )  ->  ( `' I " S )  =/=  (/) )
3937, 38syl6bi 221 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  x  e.  dom  I )  ->  ( ( I `
 x )  e.  S  ->  ( `' I " S )  =/=  (/) ) )
4039ex 425 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
x  e.  dom  I  ->  ( ( I `  x )  e.  S  ->  ( `' I " S )  =/=  (/) ) ) )
41 eleq1 2498 . . . . . . . . . . . . 13  |-  ( ( I `  x )  =  y  ->  (
( I `  x
)  e.  S  <->  y  e.  S ) )
4241biimprd 216 . . . . . . . . . . . 12  |-  ( ( I `  x )  =  y  ->  (
y  e.  S  -> 
( I `  x
)  e.  S ) )
4342imim1d 72 . . . . . . . . . . 11  |-  ( ( I `  x )  =  y  ->  (
( ( I `  x )  e.  S  ->  ( `' I " S )  =/=  (/) )  -> 
( y  e.  S  ->  ( `' I " S )  =/=  (/) ) ) )
4440, 43syl9 69 . . . . . . . . . 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 84 . . . . . . . . 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 420 . . . . . . . 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 2831 . . . . . . 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 1649 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S )  =/=  (/) )
50 eqid 2438 . . . . . 6  |-  ( glb `  K )  =  ( glb `  K )
5150, 1, 2dibglbN 32038 . . . . 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 1183 . . . 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 5748 . . . . 5  |-  ( y  e.  ( `' I " S )  ->  (
( I  |`  ( `' I " S ) ) `  y )  =  ( I `  y ) )
5453iineq2i 4114 . . . 4  |-  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^|_ y  e.  ( `' I " S ) ( I `  y
)
5552, 54syl6eqr 2488 . . 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 30230 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CLat )
5756ad2antrr 708 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  K  e.  CLat )
58 eqid 2438 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
59 eqid 2438 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
6058, 59, 1, 2dibdmN 32029 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  =  {
x  e.  ( Base `  K )  |  x ( le `  K
) W } )
61 ssrab2 3430 . . . . . . . . 9  |-  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  C_  ( Base `  K )
6260, 61syl6eqss 3400 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  C_  ( Base `  K ) )
6362adantr 453 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  dom  I  C_  ( Base `  K
) )
647, 63syl5ss 3361 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  ( Base `  K
) )
6558, 50clatglbcl 14546 . . . . . 6  |-  ( ( K  e.  CLat  /\  ( `' I " S ) 
C_  ( Base `  K
) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
6657, 64, 65syl2anc 644 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
67 n0 3639 . . . . . . 7  |-  ( ( `' I " S )  =/=  (/)  <->  E. y  y  e.  ( `' I " S ) )
6849, 67sylib 190 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  E. y 
y  e.  ( `' I " S ) )
69 hllat 30235 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
7069ad3antrrr 712 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  K  e.  Lat )
7166adantr 453 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) )  e.  ( Base `  K ) )
7264sselda 3350 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  (
Base `  K )
)
7358, 1lhpbase 30869 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7473ad3antlr 713 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  W  e.  (
Base `  K )
)
7556ad3antrrr 712 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  K  e.  CLat )
7660adantr 453 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  dom  I  =  { x  e.  ( Base `  K
)  |  x ( le `  K ) W } )
777, 76syl5sseq 3398 . . . . . . . . . 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 3362 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  ( Base `  K
) )
7978adantr 453 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( `' I " S )  C_  ( Base `  K ) )
80 simpr 449 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  ( `' I " S ) )
8158, 59, 50clatglble 14557 . . . . . . . 8  |-  ( ( K  e.  CLat  /\  ( `' I " S ) 
C_  ( Base `  K
)  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) y )
8275, 79, 80, 81syl3anc 1185 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) y )
837sseli 3346 . . . . . . . . . 10  |-  ( y  e.  ( `' I " S )  ->  y  e.  dom  I )
8483adantl 454 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  dom  I )
8558, 59, 1, 2dibeldmN 32030 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( y  e.  dom  I 
<->  ( y  e.  (
Base `  K )  /\  y ( le `  K ) W ) ) )
8685ad2antrr 708 . . . . . . . . 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 203 . . . . . . . 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 451 . . . . . . 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 14492 . . . . . 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 1649 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) ) ( le `  K
) W )
9158, 59, 1, 2dibeldmN 32030 . . . . . 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 453 . . . . 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 890 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  dom  I )
941, 2dibclN 32034 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( glb `  K ) `  ( `' I " S ) )  e.  dom  I
)  ->  ( I `  ( ( glb `  K
) `  ( `' I " S ) ) )  e.  ran  I
)
9593, 94syldan 458 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I `  ( ( glb `  K ) `  ( `' I " S ) ) )  e.  ran  I )
9655, 95eqeltrrd 2513 . 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 2513 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| S  e.  ran  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711    C_ wss 3322   (/)c0 3630   |^|cint 4052   |^|_ciin 4096   class class class wbr 4215   `'ccnv 4880   dom cdm 4881   ran crn 4882    |` cres 4883   "cima 4884   Fun wfun 5451    Fn wfn 5452   -onto->wfo 5455   -1-1-onto->wf1o 5456   ` cfv 5457   Basecbs 13474   lecple 13541   glbcglb 14405   Latclat 14479   CLatccla 14541   HLchlt 30222   LHypclh 30855   DIsoBcdib 32010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-lines 30372  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030  df-disoa 31901  df-dib 32011
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