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Metamath Proof Explorer

Theorem dicelval3

Description: Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014)

Ref Expression
Hypotheses dicval.l 
|- .<_ = ( le ` K )
dicval.a 
|- A = ( Atoms ` K )
dicval.h 
|- H = ( LHyp ` K )
dicval.p 
|- P = ( ( oc ` K ) ` W )
dicval.t 
|- T = ( ( LTrn ` K ) ` W )
dicval.e 
|- E = ( ( TEndo ` K ) ` W )
dicval.i 
|- I = ( ( DIsoC ` K ) ` W )
dicval2.g 
|- G = ( iota_ g e. T ( g ` P ) = Q )
Assertion dicelval3 
|- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> E. s e. E Y = <. ( s ` G ) , s >. ) )

Proof

Step Hyp Ref Expression
1 dicval.l 
 |-  .<_ = ( le ` K )
2 dicval.a 
 |-  A = ( Atoms ` K )
3 dicval.h 
 |-  H = ( LHyp ` K )
4 dicval.p 
 |-  P = ( ( oc ` K ) ` W )
5 dicval.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 dicval.e 
 |-  E = ( ( TEndo ` K ) ` W )
7 dicval.i 
 |-  I = ( ( DIsoC ` K ) ` W )
8 dicval2.g 
 |-  G = ( iota_ g e. T ( g ` P ) = Q )
9 1 2 3 4 5 6 7 8 dicval2 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } )
10 9 eleq2d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> Y e. { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } ) )
11 excom 
 |-  ( E. f E. s ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> E. s E. f ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) )
12 an12 
 |-  ( ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> ( f = ( s ` G ) /\ ( Y = <. f , s >. /\ s e. E ) ) )
13 12 exbii 
 |-  ( E. f ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> E. f ( f = ( s ` G ) /\ ( Y = <. f , s >. /\ s e. E ) ) )
14 fvex 
 |-  ( s ` G ) e. _V
15 opeq1 
 |-  ( f = ( s ` G ) -> <. f , s >. = <. ( s ` G ) , s >. )
16 15 eqeq2d 
 |-  ( f = ( s ` G ) -> ( Y = <. f , s >. <-> Y = <. ( s ` G ) , s >. ) )
17 16 anbi1d 
 |-  ( f = ( s ` G ) -> ( ( Y = <. f , s >. /\ s e. E ) <-> ( Y = <. ( s ` G ) , s >. /\ s e. E ) ) )
18 14 17 ceqsexv 
 |-  ( E. f ( f = ( s ` G ) /\ ( Y = <. f , s >. /\ s e. E ) ) <-> ( Y = <. ( s ` G ) , s >. /\ s e. E ) )
19 ancom 
 |-  ( ( Y = <. ( s ` G ) , s >. /\ s e. E ) <-> ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
20 13 18 19 3bitri 
 |-  ( E. f ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
21 20 exbii 
 |-  ( E. s E. f ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> E. s ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
22 11 21 bitri 
 |-  ( E. f E. s ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> E. s ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
23 elopab 
 |-  ( Y e. { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } <-> E. f E. s ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) )
24 df-rex 
 |-  ( E. s e. E Y = <. ( s ` G ) , s >. <-> E. s ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
25 22 23 24 3bitr4i 
 |-  ( Y e. { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } <-> E. s e. E Y = <. ( s ` G ) , s >. )
26 10 25 bitrdi 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> E. s e. E Y = <. ( s ` G ) , s >. ) )