Metamath Proof Explorer


Theorem resccoOLD

Description: Obsolete proof of rescco as of 14-Oct-2024. (Contributed by Mario Carneiro, 4-Jan-2017) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses rescbas.d
|- D = ( C |`cat H )
rescbas.b
|- B = ( Base ` C )
rescbas.c
|- ( ph -> C e. V )
rescbas.h
|- ( ph -> H Fn ( S X. S ) )
rescbas.s
|- ( ph -> S C_ B )
rescco.o
|- .x. = ( comp ` C )
Assertion resccoOLD
|- ( ph -> .x. = ( comp ` D ) )

Proof

Step Hyp Ref Expression
1 rescbas.d
 |-  D = ( C |`cat H )
2 rescbas.b
 |-  B = ( Base ` C )
3 rescbas.c
 |-  ( ph -> C e. V )
4 rescbas.h
 |-  ( ph -> H Fn ( S X. S ) )
5 rescbas.s
 |-  ( ph -> S C_ B )
6 rescco.o
 |-  .x. = ( comp ` C )
7 ccoid
 |-  comp = Slot ( comp ` ndx )
8 1nn0
 |-  1 e. NN0
9 4nn
 |-  4 e. NN
10 8 9 decnncl
 |-  ; 1 4 e. NN
11 10 nnrei
 |-  ; 1 4 e. RR
12 4nn0
 |-  4 e. NN0
13 5nn
 |-  5 e. NN
14 4lt5
 |-  4 < 5
15 8 12 13 14 declt
 |-  ; 1 4 < ; 1 5
16 11 15 gtneii
 |-  ; 1 5 =/= ; 1 4
17 ccondx
 |-  ( comp ` ndx ) = ; 1 5
18 homndx
 |-  ( Hom ` ndx ) = ; 1 4
19 17 18 neeq12i
 |-  ( ( comp ` ndx ) =/= ( Hom ` ndx ) <-> ; 1 5 =/= ; 1 4 )
20 16 19 mpbir
 |-  ( comp ` ndx ) =/= ( Hom ` ndx )
21 7 20 setsnid
 |-  ( comp ` ( C |`s S ) ) = ( comp ` ( ( C |`s S ) sSet <. ( Hom ` ndx ) , H >. ) )
22 2 fvexi
 |-  B e. _V
23 22 ssex
 |-  ( S C_ B -> S e. _V )
24 5 23 syl
 |-  ( ph -> S e. _V )
25 eqid
 |-  ( C |`s S ) = ( C |`s S )
26 25 6 ressco
 |-  ( S e. _V -> .x. = ( comp ` ( C |`s S ) ) )
27 24 26 syl
 |-  ( ph -> .x. = ( comp ` ( C |`s S ) ) )
28 1 3 24 4 rescval2
 |-  ( ph -> D = ( ( C |`s S ) sSet <. ( Hom ` ndx ) , H >. ) )
29 28 fveq2d
 |-  ( ph -> ( comp ` D ) = ( comp ` ( ( C |`s S ) sSet <. ( Hom ` ndx ) , H >. ) ) )
30 21 27 29 3eqtr4a
 |-  ( ph -> .x. = ( comp ` D ) )