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Theorem dibintclN 35839
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 35833 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
43adantr 465 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I : dom  I -1-1-onto-> ran  I )
5 f1ofn 5808 . . . . . 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 5348 . . . . 5  |-  ( `' I " S ) 
C_  dom  I
8 fnssres 5685 . . . . 5  |-  ( ( I  Fn  dom  I  /\  ( `' I " S )  C_  dom  I )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
96, 7, 8sylancl 662 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
10 fniinfv 5917 . . . 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 5005 . . . . 5  |-  ( I
" ( `' I " S ) )  =  ran  ( I  |`  ( `' I " S ) )
13 f1ofo 5814 . . . . . . . 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 465 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I : dom  I -onto-> ran  I
)
16 simprl 755 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  C_ 
ran  I )
17 foimacnv 5824 . . . . . 6  |-  ( ( I : dom  I -onto-> ran  I  /\  S  C_  ran  I )  ->  (
I " ( `' I " S ) )  =  S )
1815, 16, 17syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I " ( `' I " S ) )  =  S )
1912, 18syl5eqr 2515 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ran  ( I  |`  ( `' I " S ) )  =  S )
2019inteqd 4280 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| ran  ( I  |`  ( `' I " S ) )  =  |^| S
)
2111, 20eqtrd 2501 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^| S )
22 simpl 457 . . . . 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 756 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  =/=  (/) )
25 n0 3787 . . . . . . 7  |-  ( S  =/=  (/)  <->  E. y  y  e.  S )
2624, 25sylib 196 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  E. y 
y  e.  S )
2716sselda 3497 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  y  e.  ran  I
)
283ad2antrr 725 . . . . . . . . . 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 5906 . . . . . . . . 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 210 . . . . . . 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 5809 . . . . . . . . . . . . . . . 16  |-  ( I : dom  I -1-1-onto-> ran  I  ->  Fun  I )
343, 33syl 16 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
3534adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  Fun  I )
36 fvimacnv 5987 . . . . . . . . . . . . . 14  |-  ( ( Fun  I  /\  x  e.  dom  I )  -> 
( ( I `  x )  e.  S  <->  x  e.  ( `' I " S ) ) )
3735, 36sylan 471 . . . . . . . . . . . . 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 3784 . . . . . . . . . . . . 13  |-  ( x  e.  ( `' I " S )  ->  ( `' I " S )  =/=  (/) )
3937, 38syl6bi 228 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  x  e.  dom  I )  ->  ( ( I `
 x )  e.  S  ->  ( `' I " S )  =/=  (/) ) )
4039ex 434 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
x  e.  dom  I  ->  ( ( I `  x )  e.  S  ->  ( `' I " S )  =/=  (/) ) ) )
41 eleq1 2532 . . . . . . . . . . . . 13  |-  ( ( I `  x )  =  y  ->  (
( I `  x
)  e.  S  <->  y  e.  S ) )
4241biimprd 223 . . . . . . . . . . . 12  |-  ( ( I `  x )  =  y  ->  (
y  e.  S  -> 
( I `  x
)  e.  S ) )
4342imim1d 75 . . . . . . . . . . 11  |-  ( ( I `  x )  =  y  ->  (
( ( I `  x )  e.  S  ->  ( `' I " S )  =/=  (/) )  -> 
( y  e.  S  ->  ( `' I " S )  =/=  (/) ) ) )
4440, 43syl9 71 . . . . . . . . . 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 87 . . . . . . . . 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 429 . . . . . . . 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 2946 . . . . . . 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 1697 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S )  =/=  (/) )
50 eqid 2460 . . . . . 6  |-  ( glb `  K )  =  ( glb `  K )
5150, 1, 2dibglbN 35838 . . . . 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 1221 . . . 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 5871 . . . . 5  |-  ( y  e.  ( `' I " S )  ->  (
( I  |`  ( `' I " S ) ) `  y )  =  ( I `  y ) )
5453iineq2i 4338 . . . 4  |-  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^|_ y  e.  ( `' I " S ) ( I `  y
)
5552, 54syl6eqr 2519 . . 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 34030 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CLat )
5756ad2antrr 725 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  K  e.  CLat )
58 eqid 2460 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
59 eqid 2460 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
6058, 59, 1, 2dibdmN 35829 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  =  {
x  e.  ( Base `  K )  |  x ( le `  K
) W } )
61 ssrab2 3578 . . . . . . . . 9  |-  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  C_  ( Base `  K )
6260, 61syl6eqss 3547 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  C_  ( Base `  K ) )
6362adantr 465 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  dom  I  C_  ( Base `  K
) )
647, 63syl5ss 3508 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  ( Base `  K
) )
6558, 50clatglbcl 15590 . . . . . 6  |-  ( ( K  e.  CLat  /\  ( `' I " S ) 
C_  ( Base `  K
) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
6657, 64, 65syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
67 n0 3787 . . . . . . 7  |-  ( ( `' I " S )  =/=  (/)  <->  E. y  y  e.  ( `' I " S ) )
6849, 67sylib 196 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  E. y 
y  e.  ( `' I " S ) )
69 hllat 34035 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
7069ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  K  e.  Lat )
7166adantr 465 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) )  e.  ( Base `  K ) )
7264sselda 3497 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  (
Base `  K )
)
7358, 1lhpbase 34669 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7473ad3antlr 730 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  W  e.  (
Base `  K )
)
7556ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  K  e.  CLat )
7660adantr 465 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  dom  I  =  { x  e.  ( Base `  K
)  |  x ( le `  K ) W } )
777, 76syl5sseq 3545 . . . . . . . . . 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 3509 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  ( Base `  K
) )
7978adantr 465 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( `' I " S )  C_  ( Base `  K ) )
80 simpr 461 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  ( `' I " S ) )
8158, 59, 50clatglble 15601 . . . . . . . 8  |-  ( ( K  e.  CLat  /\  ( `' I " S ) 
C_  ( Base `  K
)  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) y )
8275, 79, 80, 81syl3anc 1223 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) y )
837sseli 3493 . . . . . . . . . 10  |-  ( y  e.  ( `' I " S )  ->  y  e.  dom  I )
8483adantl 466 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  dom  I )
8558, 59, 1, 2dibeldmN 35830 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( y  e.  dom  I 
<->  ( y  e.  (
Base `  K )  /\  y ( le `  K ) W ) ) )
8685ad2antrr 725 . . . . . . . . 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 210 . . . . . . . 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 463 . . . . . . 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 15534 . . . . . 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 1697 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) ) ( le `  K
) W )
9158, 59, 1, 2dibeldmN 35830 . . . . . 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 465 . . . . 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 915 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  dom  I )
941, 2dibclN 35834 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( glb `  K ) `  ( `' I " S ) )  e.  dom  I
)  ->  ( I `  ( ( glb `  K
) `  ( `' I " S ) ) )  e.  ran  I
)
9593, 94syldan 470 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I `  ( ( glb `  K ) `  ( `' I " S ) ) )  e.  ran  I )
9655, 95eqeltrrd 2549 . 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 2549 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 184    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2655   E.wrex 2808   {crab 2811    C_ wss 3469   (/)c0 3778   |^|cint 4275   |^|_ciin 4319   class class class wbr 4440   `'ccnv 4991   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995   Fun wfun 5573    Fn wfn 5574   -onto->wfo 5577   -1-1-onto->wf1o 5578   ` cfv 5579   Basecbs 14479   lecple 14551   glbcglb 15419   Latclat 15521   CLatccla 15583   HLchlt 34022   LHypclh 34655   DIsoBcdib 35810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-undef 6992  df-map 7412  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830  df-disoa 35701  df-dib 35811
This theorem is referenced by: (None)
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