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Theorem diag12 15069
Description: Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
diagval.l  |-  L  =  ( CΔfunc D )
diagval.c  |-  ( ph  ->  C  e.  Cat )
diagval.d  |-  ( ph  ->  D  e.  Cat )
diag11.a  |-  A  =  ( Base `  C
)
diag11.c  |-  ( ph  ->  X  e.  A )
diag11.k  |-  K  =  ( ( 1st `  L
) `  X )
diag11.b  |-  B  =  ( Base `  D
)
diag11.y  |-  ( ph  ->  Y  e.  B )
diag12.j  |-  J  =  ( Hom  `  D
)
diag12.i  |-  .1.  =  ( Id `  C )
diag12.z  |-  ( ph  ->  Z  e.  B )
diag12.f  |-  ( ph  ->  F  e.  ( Y J Z ) )
Assertion
Ref Expression
diag12  |-  ( ph  ->  ( ( Y ( 2nd `  K ) Z ) `  F
)  =  (  .1.  `  X ) )

Proof of Theorem diag12
StepHypRef Expression
1 diag11.k . . . . . 6  |-  K  =  ( ( 1st `  L
) `  X )
2 diagval.l . . . . . . . . 9  |-  L  =  ( CΔfunc D )
3 diagval.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
4 diagval.d . . . . . . . . 9  |-  ( ph  ->  D  e.  Cat )
52, 3, 4diagval 15065 . . . . . . . 8  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
65fveq2d 5710 . . . . . . 7  |-  ( ph  ->  ( 1st `  L
)  =  ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) )
76fveq1d 5708 . . . . . 6  |-  ( ph  ->  ( ( 1st `  L
) `  X )  =  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) )
81, 7syl5eq 2487 . . . . 5  |-  ( ph  ->  K  =  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) )
98fveq2d 5710 . . . 4  |-  ( ph  ->  ( 2nd `  K
)  =  ( 2nd `  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) ) )
109oveqd 6123 . . 3  |-  ( ph  ->  ( Y ( 2nd `  K ) Z )  =  ( Y ( 2nd `  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) ) Z ) )
1110fveq1d 5708 . 2  |-  ( ph  ->  ( ( Y ( 2nd `  K ) Z ) `  F
)  =  ( ( Y ( 2nd `  (
( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) ) Z ) `  F
) )
12 eqid 2443 . . 3  |-  ( <. C ,  D >. curryF  ( C  1stF  D ) )  =  ( <. C ,  D >. curryF  ( C  1stF  D ) )
13 diag11.a . . 3  |-  A  =  ( Base `  C
)
14 eqid 2443 . . . 4  |-  ( C  X.c  D )  =  ( C  X.c  D )
15 eqid 2443 . . . 4  |-  ( C  1stF  D )  =  ( C  1stF  D )
1614, 3, 4, 151stfcl 15022 . . 3  |-  ( ph  ->  ( C  1stF  D )  e.  ( ( C  X.c  D
)  Func  C )
)
17 diag11.b . . 3  |-  B  =  ( Base `  D
)
18 diag11.c . . 3  |-  ( ph  ->  X  e.  A )
19 eqid 2443 . . 3  |-  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X )  =  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X )
20 diag11.y . . 3  |-  ( ph  ->  Y  e.  B )
21 diag12.j . . 3  |-  J  =  ( Hom  `  D
)
22 diag12.i . . 3  |-  .1.  =  ( Id `  C )
23 diag12.z . . 3  |-  ( ph  ->  Z  e.  B )
24 diag12.f . . 3  |-  ( ph  ->  F  e.  ( Y J Z ) )
2512, 13, 3, 4, 16, 17, 18, 19, 20, 21, 22, 23, 24curf12 15052 . 2  |-  ( ph  ->  ( ( Y ( 2nd `  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) ) Z ) `  F
)  =  ( (  .1.  `  X )
( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. X ,  Z >. ) F ) )
26 df-ov 6109 . . . 4  |-  ( (  .1.  `  X )
( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. X ,  Z >. ) F )  =  ( ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. X ,  Z >. ) `
 <. (  .1.  `  X ) ,  F >. )
2714, 13, 17xpcbas 15003 . . . . . 6  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
28 eqid 2443 . . . . . 6  |-  ( Hom  `  ( C  X.c  D ) )  =  ( Hom  `  ( C  X.c  D ) )
29 opelxpi 4886 . . . . . . 7  |-  ( ( X  e.  A  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
3018, 20, 29syl2anc 661 . . . . . 6  |-  ( ph  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
31 opelxpi 4886 . . . . . . 7  |-  ( ( X  e.  A  /\  Z  e.  B )  -> 
<. X ,  Z >.  e.  ( A  X.  B
) )
3218, 23, 31syl2anc 661 . . . . . 6  |-  ( ph  -> 
<. X ,  Z >.  e.  ( A  X.  B
) )
3314, 27, 28, 3, 4, 15, 30, 321stf2 15018 . . . . 5  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. X ,  Z >. )  =  ( 1st  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. X ,  Z >. ) ) )
3433fveq1d 5708 . . . 4  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. X ,  Z >. ) `
 <. (  .1.  `  X ) ,  F >. )  =  ( ( 1st  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D
) ) <. X ,  Z >. ) ) `  <. (  .1.  `  X
) ,  F >. ) )
3526, 34syl5eq 2487 . . 3  |-  ( ph  ->  ( (  .1.  `  X ) ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. X ,  Z >. ) F )  =  ( ( 1st  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. X ,  Z >. ) ) `  <. (  .1.  `  X ) ,  F >. ) )
36 eqid 2443 . . . . . . 7  |-  ( Hom  `  C )  =  ( Hom  `  C )
3713, 36, 22, 3, 18catidcl 14635 . . . . . 6  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X ( Hom  `  C
) X ) )
38 opelxpi 4886 . . . . . 6  |-  ( ( (  .1.  `  X
)  e.  ( X ( Hom  `  C
) X )  /\  F  e.  ( Y J Z ) )  ->  <. (  .1.  `  X
) ,  F >.  e.  ( ( X ( Hom  `  C ) X )  X.  ( Y J Z ) ) )
3937, 24, 38syl2anc 661 . . . . 5  |-  ( ph  -> 
<. (  .1.  `  X
) ,  F >.  e.  ( ( X ( Hom  `  C ) X )  X.  ( Y J Z ) ) )
4014, 13, 17, 36, 21, 18, 20, 18, 23, 28xpchom2 15011 . . . . 5  |-  ( ph  ->  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. X ,  Z >. )  =  ( ( X ( Hom  `  C
) X )  X.  ( Y J Z ) ) )
4139, 40eleqtrrd 2520 . . . 4  |-  ( ph  -> 
<. (  .1.  `  X
) ,  F >.  e.  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. X ,  Z >. ) )
42 fvres 5719 . . . 4  |-  ( <.
(  .1.  `  X
) ,  F >.  e.  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. X ,  Z >. )  ->  ( ( 1st  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. X ,  Z >. ) ) `  <. (  .1.  `  X ) ,  F >. )  =  ( 1st `  <. (  .1.  `  X ) ,  F >. ) )
4341, 42syl 16 . . 3  |-  ( ph  ->  ( ( 1st  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. X ,  Z >. ) ) `  <. (  .1.  `  X ) ,  F >. )  =  ( 1st `  <. (  .1.  `  X ) ,  F >. ) )
44 op1stg 6604 . . . 4  |-  ( ( (  .1.  `  X
)  e.  ( X ( Hom  `  C
) X )  /\  F  e.  ( Y J Z ) )  -> 
( 1st `  <. (  .1.  `  X ) ,  F >. )  =  (  .1.  `  X )
)
4537, 24, 44syl2anc 661 . . 3  |-  ( ph  ->  ( 1st `  <. (  .1.  `  X ) ,  F >. )  =  (  .1.  `  X )
)
4635, 43, 453eqtrd 2479 . 2  |-  ( ph  ->  ( (  .1.  `  X ) ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. X ,  Z >. ) F )  =  (  .1.  `  X )
)
4711, 25, 463eqtrd 2479 1  |-  ( ph  ->  ( ( Y ( 2nd `  K ) Z ) `  F
)  =  (  .1.  `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   <.cop 3898    X. cxp 4853    |` cres 4857   ` cfv 5433  (class class class)co 6106   1stc1st 6590   2ndc2nd 6591   Basecbs 14189   Hom chom 14264   Catccat 14617   Idccid 14618    X.c cxpc 14993    1stF c1stf 14994   curryF ccurf 15035  Δfunccdiag 15037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-ixp 7279  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-fz 11453  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-hom 14277  df-cco 14278  df-cat 14621  df-cid 14622  df-func 14783  df-xpc 14997  df-1stf 14998  df-curf 15039  df-diag 15041
This theorem is referenced by:  curf2ndf  15072
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