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Theorem diag2 14035
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 14030 . . . . 5  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
54fveq2d 5545 . . . 4  |-  ( ph  ->  ( 2nd `  L
)  =  ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) )
65oveqd 5891 . . 3  |-  ( ph  ->  ( X ( 2nd `  L ) Y )  =  ( X ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) Y ) )
76fveq1d 5543 . 2  |-  ( ph  ->  ( ( X ( 2nd `  L ) Y ) `  F
)  =  ( ( X ( 2nd `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) Y ) `  F
) )
8 eqid 2296 . . 3  |-  ( <. C ,  D >. curryF  ( C  1stF  D ) )  =  ( <. C ,  D >. curryF  ( C  1stF  D ) )
9 diag2.a . . 3  |-  A  =  ( Base `  C
)
10 eqid 2296 . . . 4  |-  ( C  X.c  D )  =  ( C  X.c  D )
11 eqid 2296 . . . 4  |-  ( C  1stF  D )  =  ( C  1stF  D )
1210, 2, 3, 111stfcl 13987 . . 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 2296 . . 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 2296 . . 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 14019 . 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 13968 . . . . . . 7  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
22 eqid 2296 . . . . . . 7  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
232adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  C  e.  Cat )
243adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  D  e.  Cat )
25 opelxpi 4737 . . . . . . . 8  |-  ( ( X  e.  A  /\  x  e.  B )  -> 
<. X ,  x >.  e.  ( A  X.  B
) )
2616, 25sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  <. X ,  x >.  e.  ( A  X.  B ) )
27 opelxpi 4737 . . . . . . . 8  |-  ( ( Y  e.  A  /\  x  e.  B )  -> 
<. Y ,  x >.  e.  ( A  X.  B
) )
2817, 27sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  <. Y ,  x >.  e.  ( A  X.  B ) )
2910, 21, 22, 23, 24, 11, 26, 281stf2 13983 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. )  =  ( 1st  |`  ( <. X ,  x >. (  Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) ) )
3029oveqd 5891 . . . . 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 5877 . . . . . 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 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  F  e.  ( X H Y ) )
33 eqid 2296 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
34 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
3513, 33, 15, 24, 34catidcl 13600 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  (
( Id `  D
) `  x )  e.  ( x (  Hom  `  D ) x ) )
36 opelxpi 4737 . . . . . . . . 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 642 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  <. F , 
( ( Id `  D ) `  x
) >.  e.  ( ( X H Y )  X.  ( x (  Hom  `  D )
x ) ) )
3816adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  X  e.  A )
3917adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  Y  e.  A )
4010, 9, 13, 14, 33, 38, 34, 39, 34, 22xpchom2 13976 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( <. X ,  x >. (  Hom  `  ( C  X.c  D ) ) <. Y ,  x >. )  =  ( ( X H Y )  X.  ( x (  Hom  `  D ) x ) ) )
4137, 40eleqtrrd 2373 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  <. F , 
( ( Id `  D ) `  x
) >.  e.  ( <. X ,  x >. (  Hom  `  ( C  X.c  D ) ) <. Y ,  x >. ) )
42 fvres 5558 . . . . . . 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 15 . . . . . 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 2340 . . . . 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 6148 . . . . . 6  |-  ( ( F  e.  ( X H Y )  /\  ( ( Id `  D ) `  x
)  e.  ( x (  Hom  `  D
) x ) )  ->  ( 1st `  <. F ,  ( ( Id
`  D ) `  x ) >. )  =  F )
4632, 35, 45syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( 1st `  <. F ,  ( ( Id `  D
) `  x ) >. )  =  F )
4730, 44, 463eqtrd 2332 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) )  =  F )
4847mpteq2dva 4122 . . 3  |-  ( ph  ->  ( x  e.  B  |->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) ) )  =  ( x  e.  B  |->  F ) )
49 fconstmpt 4748 . . 3  |-  ( B  X.  { F }
)  =  ( x  e.  B  |->  F )
5048, 49syl6eqr 2346 . 2  |-  ( ph  ->  ( x  e.  B  |->  ( F ( <. X ,  x >. ( 2nd `  ( C  1stF  D ) ) <. Y ,  x >. ) ( ( Id `  D ) `  x
) ) )  =  ( B  X.  { F } ) )
517, 20, 503eqtrd 2332 1  |-  ( ph  ->  ( ( X ( 2nd `  L ) Y ) `  F
)  =  ( B  X.  { F }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   <.cop 3656    e. cmpt 4093    X. cxp 4703    |` cres 4707   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235   Catccat 13582   Idccid 13583    X.c cxpc 13958    1stF c1stf 13959   curryF ccurf 14000  Δfunccdiag 14002
This theorem is referenced by:  diag2cl  14036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-func 13748  df-xpc 13962  df-1stf 13963  df-curf 14004  df-diag 14006
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