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Theorem diag2 15385
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 15380 . . . . 5  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
54fveq2d 5857 . . . 4  |-  ( ph  ->  ( 2nd `  L
)  =  ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) )
65oveqd 6295 . . 3  |-  ( ph  ->  ( X ( 2nd `  L ) Y )  =  ( X ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) Y ) )
76fveq1d 5855 . 2  |-  ( ph  ->  ( ( X ( 2nd `  L ) Y ) `  F
)  =  ( ( X ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) Y ) `  F
) )
8 eqid 2441 . . 3  |-  ( <. C ,  D >. curryF  ( C  1stF  D ) )  =  ( <. C ,  D >. curryF  ( C  1stF  D ) )
9 diag2.a . . 3  |-  A  =  ( Base `  C
)
10 eqid 2441 . . . 4  |-  ( C  X.c  D )  =  ( C  X.c  D )
11 eqid 2441 . . . 4  |-  ( C  1stF  D )  =  ( C  1stF  D )
1210, 2, 3, 111stfcl 15337 . . 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 2441 . . 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 2441 . . 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 15369 . 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 15318 . . . . . . 7  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
22 eqid 2441 . . . . . . 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 5018 . . . . . . . 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 5018 . . . . . . . 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 15333 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. )  =  ( 1st  |`  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) ) )
3029oveqd 6295 . . . . 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 6281 . . . . . 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 2441 . . . . . . . . . 10  |-  ( Hom  `  D )  =  ( Hom  `  D )
34 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
3513, 33, 15, 24, 34catidcl 14953 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  (
( Id `  D
) `  x )  e.  ( x ( Hom  `  D ) x ) )
36 opelxpi 5018 . . . . . . . . 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 15326 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. )  =  ( ( X H Y )  X.  ( x ( Hom  `  D ) x ) ) )
4137, 40eleqtrrd 2532 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  <. F , 
( ( Id `  D ) `  x
) >.  e.  ( <. X ,  x >. ( Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) )
42 fvres 5867 . . . . . . 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 2494 . . . . 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 6794 . . . . . 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 2486 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) )  =  F )
4847mpteq2dva 4520 . . 3  |-  ( ph  ->  ( x  e.  B  |->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) ) )  =  ( x  e.  B  |->  F ) )
49 fconstmpt 5030 . . 3  |-  ( B  X.  { F }
)  =  ( x  e.  B  |->  F )
5048, 49syl6eqr 2500 . 2  |-  ( ph  ->  ( x  e.  B  |->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) ) )  =  ( B  X.  { F } ) )
517, 20, 503eqtrd 2486 1  |-  ( ph  ->  ( ( X ( 2nd `  L ) Y ) `  F
)  =  ( B  X.  { F }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   {csn 4011   <.cop 4017    |-> cmpt 4492    X. cxp 4984    |` cres 4988   ` cfv 5575  (class class class)co 6278   1stc1st 6780   2ndc2nd 6781   Basecbs 14506   Hom chom 14582   Catccat 14935   Idccid 14936    X.c cxpc 15308    1stF c1stf 15309   curryF ccurf 15350  Δfunccdiag 15352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-map 7421  df-ixp 7469  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-10 10605  df-n0 10799  df-z 10868  df-dec 10982  df-uz 11088  df-fz 11679  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-hom 14595  df-cco 14596  df-cat 14939  df-cid 14940  df-func 15098  df-xpc 15312  df-1stf 15313  df-curf 15354  df-diag 15356
This theorem is referenced by:  diag2cl  15386
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