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Theorem diag2 15055
Description: Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
diag2.l  |-  L  =  ( CΔfunc D )
diag2.a  |-  A  =  ( Base `  C
)
diag2.b  |-  B  =  ( Base `  D
)
diag2.h  |-  H  =  ( Hom  `  C
)
diag2.c  |-  ( ph  ->  C  e.  Cat )
diag2.d  |-  ( ph  ->  D  e.  Cat )
diag2.x  |-  ( ph  ->  X  e.  A )
diag2.y  |-  ( ph  ->  Y  e.  A )
diag2.f  |-  ( ph  ->  F  e.  ( X H Y ) )
Assertion
Ref Expression
diag2  |-  ( ph  ->  ( ( X ( 2nd `  L ) Y ) `  F
)  =  ( B  X.  { F }
) )

Proof of Theorem diag2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 diag2.l . . . . . 6  |-  L  =  ( CΔfunc D )
2 diag2.c . . . . . 6  |-  ( ph  ->  C  e.  Cat )
3 diag2.d . . . . . 6  |-  ( ph  ->  D  e.  Cat )
41, 2, 3diagval 15050 . . . . 5  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
54fveq2d 5695 . . . 4  |-  ( ph  ->  ( 2nd `  L
)  =  ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) )
65oveqd 6108 . . 3  |-  ( ph  ->  ( X ( 2nd `  L ) Y )  =  ( X ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) Y ) )
76fveq1d 5693 . 2  |-  ( ph  ->  ( ( X ( 2nd `  L ) Y ) `  F
)  =  ( ( X ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) Y ) `  F
) )
8 eqid 2443 . . 3  |-  ( <. C ,  D >. curryF  ( C  1stF  D ) )  =  ( <. C ,  D >. curryF  ( C  1stF  D ) )
9 diag2.a . . 3  |-  A  =  ( Base `  C
)
10 eqid 2443 . . . 4  |-  ( C  X.c  D )  =  ( C  X.c  D )
11 eqid 2443 . . . 4  |-  ( C  1stF  D )  =  ( C  1stF  D )
1210, 2, 3, 111stfcl 15007 . . 3  |-  ( ph  ->  ( C  1stF  D )  e.  ( ( C  X.c  D
)  Func  C )
)
13 diag2.b . . 3  |-  B  =  ( Base `  D
)
14 diag2.h . . 3  |-  H  =  ( Hom  `  C
)
15 eqid 2443 . . 3  |-  ( Id
`  D )  =  ( Id `  D
)
16 diag2.x . . 3  |-  ( ph  ->  X  e.  A )
17 diag2.y . . 3  |-  ( ph  ->  Y  e.  A )
18 diag2.f . . 3  |-  ( ph  ->  F  e.  ( X H Y ) )
19 eqid 2443 . . 3  |-  ( ( X ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) Y ) `  F
)  =  ( ( X ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) Y ) `  F
)
208, 9, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19curf2 15039 . 2  |-  ( ph  ->  ( ( X ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) Y ) `  F
)  =  ( x  e.  B  |->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) ) ) )
2110, 9, 13xpcbas 14988 . . . . . . 7  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
22 eqid 2443 . . . . . . 7  |-  ( Hom  `  ( C  X.c  D ) )  =  ( Hom  `  ( C  X.c  D ) )
232adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  C  e.  Cat )
243adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  D  e.  Cat )
25 opelxpi 4871 . . . . . . . 8  |-  ( ( X  e.  A  /\  x  e.  B )  -> 
<. X ,  x >.  e.  ( A  X.  B
) )
2616, 25sylan 471 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  <. X ,  x >.  e.  ( A  X.  B ) )
27 opelxpi 4871 . . . . . . . 8  |-  ( ( Y  e.  A  /\  x  e.  B )  -> 
<. Y ,  x >.  e.  ( A  X.  B
) )
2817, 27sylan 471 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  <. Y ,  x >.  e.  ( A  X.  B ) )
2910, 21, 22, 23, 24, 11, 26, 281stf2 15003 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. )  =  ( 1st  |`  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) ) )
3029oveqd 6108 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) )  =  ( F ( 1st  |`  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) ) ( ( Id
`  D ) `  x ) ) )
31 df-ov 6094 . . . . . 6  |-  ( F ( 1st  |`  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) ) ( ( Id
`  D ) `  x ) )  =  ( ( 1st  |`  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) ) `  <. F , 
( ( Id `  D ) `  x
) >. )
3218adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  F  e.  ( X H Y ) )
33 eqid 2443 . . . . . . . . . 10  |-  ( Hom  `  D )  =  ( Hom  `  D )
34 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
3513, 33, 15, 24, 34catidcl 14620 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  (
( Id `  D
) `  x )  e.  ( x ( Hom  `  D ) x ) )
36 opelxpi 4871 . . . . . . . . 9  |-  ( ( F  e.  ( X H Y )  /\  ( ( Id `  D ) `  x
)  e.  ( x ( Hom  `  D
) x ) )  ->  <. F ,  ( ( Id `  D
) `  x ) >.  e.  ( ( X H Y )  X.  ( x ( Hom  `  D ) x ) ) )
3732, 35, 36syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  <. F , 
( ( Id `  D ) `  x
) >.  e.  ( ( X H Y )  X.  ( x ( Hom  `  D )
x ) ) )
3816adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  X  e.  A )
3917adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  Y  e.  A )
4010, 9, 13, 14, 33, 38, 34, 39, 34, 22xpchom2 14996 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. )  =  ( ( X H Y )  X.  ( x ( Hom  `  D ) x ) ) )
4137, 40eleqtrrd 2520 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  <. F , 
( ( Id `  D ) `  x
) >.  e.  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) )
42 fvres 5704 . . . . . . 7  |-  ( <. F ,  ( ( Id `  D ) `  x ) >.  e.  (
<. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. )  ->  ( ( 1st  |`  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) ) `  <. F , 
( ( Id `  D ) `  x
) >. )  =  ( 1st `  <. F , 
( ( Id `  D ) `  x
) >. ) )
4341, 42syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st  |`  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) ) `  <. F , 
( ( Id `  D ) `  x
) >. )  =  ( 1st `  <. F , 
( ( Id `  D ) `  x
) >. ) )
4431, 43syl5eq 2487 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( F ( 1st  |`  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) ) ( ( Id
`  D ) `  x ) )  =  ( 1st `  <. F ,  ( ( Id
`  D ) `  x ) >. )
)
45 op1stg 6589 . . . . . 6  |-  ( ( F  e.  ( X H Y )  /\  ( ( Id `  D ) `  x
)  e.  ( x ( Hom  `  D
) x ) )  ->  ( 1st `  <. F ,  ( ( Id
`  D ) `  x ) >. )  =  F )
4632, 35, 45syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( 1st `  <. F ,  ( ( Id `  D
) `  x ) >. )  =  F )
4730, 44, 463eqtrd 2479 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) )  =  F )
4847mpteq2dva 4378 . . 3  |-  ( ph  ->  ( x  e.  B  |->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) ) )  =  ( x  e.  B  |->  F ) )
49 fconstmpt 4882 . . 3  |-  ( B  X.  { F }
)  =  ( x  e.  B  |->  F )
5048, 49syl6eqr 2493 . 2  |-  ( ph  ->  ( x  e.  B  |->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) ) )  =  ( B  X.  { F } ) )
517, 20, 503eqtrd 2479 1  |-  ( ph  ->  ( ( X ( 2nd `  L ) Y ) `  F
)  =  ( B  X.  { F }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3877   <.cop 3883    e. cmpt 4350    X. cxp 4838    |` cres 4842   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   Basecbs 14174   Hom chom 14249   Catccat 14602   Idccid 14603    X.c cxpc 14978    1stF c1stf 14979   curryF ccurf 15020  Δfunccdiag 15022
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-hom 14262  df-cco 14263  df-cat 14606  df-cid 14607  df-func 14768  df-xpc 14982  df-1stf 14983  df-curf 15024  df-diag 15026
This theorem is referenced by:  diag2cl  15056
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