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Theorem diaintclN 34364
Description: The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diaintcl.h  |-  H  =  ( LHyp `  K
)
diaintcl.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diaintclN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| S  e.  ran  I )

Proof of Theorem diaintclN
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 diaintcl.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
2 diaintcl.i . . . . . . . 8  |-  I  =  ( ( DIsoA `  K
) `  W )
31, 2diaf11N 34355 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
43adantr 466 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I : dom  I -1-1-onto-> ran  I )
5 f1ofn 5823 . . . . . 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 5199 . . . . 5  |-  ( `' I " S ) 
C_  dom  I
8 fnssres 5698 . . . . 5  |-  ( ( I  Fn  dom  I  /\  ( `' I " S )  C_  dom  I )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
96, 7, 8sylancl 666 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
10 fniinfv 5931 . . . 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 4858 . . . . 5  |-  ( I
" ( `' I " S ) )  =  ran  ( I  |`  ( `' I " S ) )
13 f1ofo 5829 . . . . . . . 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 466 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I : dom  I -onto-> ran  I
)
16 simprl 762 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  C_ 
ran  I )
17 foimacnv 5839 . . . . . 6  |-  ( ( I : dom  I -onto-> ran  I  /\  S  C_  ran  I )  ->  (
I " ( `' I " S ) )  =  S )
1815, 16, 17syl2anc 665 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I " ( `' I " S ) )  =  S )
1912, 18syl5eqr 2475 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ran  ( I  |`  ( `' I " S ) )  =  S )
2019inteqd 4254 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| ran  ( I  |`  ( `' I " S ) )  =  |^| S
)
2111, 20eqtrd 2461 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^| S )
22 simpl 458 . . . . 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 764 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  =/=  (/) )
25 n0 3768 . . . . . . 7  |-  ( S  =/=  (/)  <->  E. y  y  e.  S )
2624, 25sylib 199 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  E. y 
y  e.  S )
2716sselda 3461 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  y  e.  ran  I
)
283ad2antrr 730 . . . . . . . . . 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 5919 . . . . . . . . 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 213 . . . . . . 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 5824 . . . . . . . . . . . . . . . 16  |-  ( I : dom  I -1-1-onto-> ran  I  ->  Fun  I )
343, 33syl 17 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
3534adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  Fun  I )
36 fvimacnv 6003 . . . . . . . . . . . . . 14  |-  ( ( Fun  I  /\  x  e.  dom  I )  -> 
( ( I `  x )  e.  S  <->  x  e.  ( `' I " S ) ) )
3735, 36sylan 473 . . . . . . . . . . . . 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 3764 . . . . . . . . . . . . 13  |-  ( x  e.  ( `' I " S )  ->  ( `' I " S )  =/=  (/) )
3937, 38syl6bi 231 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  x  e.  dom  I )  ->  ( ( I `
 x )  e.  S  ->  ( `' I " S )  =/=  (/) ) )
4039ex 435 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
x  e.  dom  I  ->  ( ( I `  x )  e.  S  ->  ( `' I " S )  =/=  (/) ) ) )
41 eleq1 2492 . . . . . . . . . . . . 13  |-  ( ( I `  x )  =  y  ->  (
( I `  x
)  e.  S  <->  y  e.  S ) )
4241biimprd 226 . . . . . . . . . . . 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 430 . . . . . . . 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 2913 . . . . . . 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 1770 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S )  =/=  (/) )
50 eqid 2420 . . . . . 6  |-  ( glb `  K )  =  ( glb `  K )
5150, 1, 2diaglbN 34361 . . . . 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 1262 . . . 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 5886 . . . . 5  |-  ( y  e.  ( `' I " S )  ->  (
( I  |`  ( `' I " S ) ) `  y )  =  ( I `  y ) )
5453iineq2i 4313 . . . 4  |-  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^|_ y  e.  ( `' I " S ) ( I `  y
)
5552, 54syl6eqr 2479 . . 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 32662 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CLat )
5756ad2antrr 730 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  K  e.  CLat )
58 eqid 2420 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
59 eqid 2420 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
6058, 59, 1, 2diadm 34341 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  =  {
x  e.  ( Base `  K )  |  x ( le `  K
) W } )
61 ssrab2 3543 . . . . . . . . 9  |-  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  C_  ( Base `  K )
6260, 61syl6eqss 3511 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  C_  ( Base `  K ) )
6362adantr 466 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  dom  I  C_  ( Base `  K
) )
647, 63syl5ss 3472 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  ( Base `  K
) )
6558, 50clatglbcl 16304 . . . . . 6  |-  ( ( K  e.  CLat  /\  ( `' I " S ) 
C_  ( Base `  K
) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
6657, 64, 65syl2anc 665 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
67 n0 3768 . . . . . . 7  |-  ( ( `' I " S )  =/=  (/)  <->  E. y  y  e.  ( `' I " S ) )
6849, 67sylib 199 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  E. y 
y  e.  ( `' I " S ) )
69 hllat 32667 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
7069ad3antrrr 734 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  K  e.  Lat )
7166adantr 466 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) )  e.  ( Base `  K ) )
7264sselda 3461 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  (
Base `  K )
)
7358, 1lhpbase 33301 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7473ad3antlr 735 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  W  e.  (
Base `  K )
)
7556ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  K  e.  CLat )
7660adantr 466 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  dom  I  =  { x  e.  ( Base `  K
)  |  x ( le `  K ) W } )
777, 76syl5sseq 3509 . . . . . . . . . 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 3473 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  ( Base `  K
) )
7978adantr 466 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( `' I " S )  C_  ( Base `  K ) )
80 simpr 462 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  ( `' I " S ) )
8158, 59, 50clatglble 16315 . . . . . . . 8  |-  ( ( K  e.  CLat  /\  ( `' I " S ) 
C_  ( Base `  K
)  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) y )
8275, 79, 80, 81syl3anc 1264 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  ( ( glb `  K ) `  ( `' I " S ) ) ( le `  K ) y )
837sseli 3457 . . . . . . . . . 10  |-  ( y  e.  ( `' I " S )  ->  y  e.  dom  I )
8483adantl 467 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  ( `' I " S ) )  ->  y  e.  dom  I )
8558, 59, 1, 2diaeldm 34342 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( y  e.  dom  I 
<->  ( y  e.  (
Base `  K )  /\  y ( le `  K ) W ) ) )
8685ad2antrr 730 . . . . . . . . 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 213 . . . . . . . 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 464 . . . . . . 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 16248 . . . . . 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 1770 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) ) ( le `  K
) W )
9158, 59, 1, 2diaeldm 34342 . . . . . 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 466 . . . . 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 930 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  dom  I )
941, 2diaclN 34356 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( glb `  K ) `  ( `' I " S ) )  e.  dom  I
)  ->  ( I `  ( ( glb `  K
) `  ( `' I " S ) ) )  e.  ran  I
)
9593, 94syldan 472 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I `  ( ( glb `  K ) `  ( `' I " S ) ) )  e.  ran  I )
9655, 95eqeltrrd 2509 . 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 2509 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 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867    =/= wne 2616   E.wrex 2774   {crab 2777    C_ wss 3433   (/)c0 3758   |^|cint 4249   |^|_ciin 4294   class class class wbr 4417   `'ccnv 4844   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848   Fun wfun 5586    Fn wfn 5587   -onto->wfo 5590   -1-1-onto->wf1o 5591   ` cfv 5592   Basecbs 15073   lecple 15149   glbcglb 16132   Latclat 16235   CLatccla 16297   HLchlt 32654   LHypclh 33287   DIsoAcdia 34334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-riotaBAD 32263
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-undef 7019  df-map 7473  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-p0 16229  df-p1 16230  df-lat 16236  df-clat 16298  df-oposet 32480  df-ol 32482  df-oml 32483  df-covers 32570  df-ats 32571  df-atl 32602  df-cvlat 32626  df-hlat 32655  df-llines 32801  df-lplanes 32802  df-lvols 32803  df-lines 32804  df-psubsp 32806  df-pmap 32807  df-padd 33099  df-lhyp 33291  df-laut 33292  df-ldil 33407  df-ltrn 33408  df-trl 33463  df-disoa 34335
This theorem is referenced by:  docaclN  34430
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