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Theorem diag11 15152
Description: Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-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 )
Assertion
Ref Expression
diag11  |-  ( ph  ->  ( ( 1st `  K
) `  Y )  =  X )

Proof of Theorem diag11
StepHypRef Expression
1 diag11.k . . . . 5  |-  K  =  ( ( 1st `  L
) `  X )
2 diagval.l . . . . . . . 8  |-  L  =  ( CΔfunc D )
3 diagval.c . . . . . . . 8  |-  ( ph  ->  C  e.  Cat )
4 diagval.d . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
52, 3, 4diagval 15149 . . . . . . 7  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
65fveq2d 5790 . . . . . 6  |-  ( ph  ->  ( 1st `  L
)  =  ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) )
76fveq1d 5788 . . . . 5  |-  ( ph  ->  ( ( 1st `  L
) `  X )  =  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) )
81, 7syl5eq 2503 . . . 4  |-  ( ph  ->  K  =  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) )
98fveq2d 5790 . . 3  |-  ( ph  ->  ( 1st `  K
)  =  ( 1st `  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) ) )
109fveq1d 5788 . 2  |-  ( ph  ->  ( ( 1st `  K
) `  Y )  =  ( ( 1st `  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) ) `
 Y ) )
11 eqid 2451 . . 3  |-  ( <. C ,  D >. curryF  ( C  1stF  D ) )  =  ( <. C ,  D >. curryF  ( C  1stF  D ) )
12 diag11.a . . 3  |-  A  =  ( Base `  C
)
13 eqid 2451 . . . 4  |-  ( C  X.c  D )  =  ( C  X.c  D )
14 eqid 2451 . . . 4  |-  ( C  1stF  D )  =  ( C  1stF  D )
1513, 3, 4, 141stfcl 15106 . . 3  |-  ( ph  ->  ( C  1stF  D )  e.  ( ( C  X.c  D
)  Func  C )
)
16 diag11.b . . 3  |-  B  =  ( Base `  D
)
17 diag11.c . . 3  |-  ( ph  ->  X  e.  A )
18 eqid 2451 . . 3  |-  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X )  =  ( ( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X )
19 diag11.y . . 3  |-  ( ph  ->  Y  e.  B )
2011, 12, 3, 4, 15, 16, 17, 18, 19curf11 15135 . 2  |-  ( ph  ->  ( ( 1st `  (
( 1st `  ( <. C ,  D >. curryF  ( C  1stF  D ) ) ) `
 X ) ) `
 Y )  =  ( X ( 1st `  ( C  1stF  D )
) Y ) )
21 df-ov 6190 . . . 4  |-  ( X ( 1st `  ( C  1stF  D ) ) Y )  =  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )
2213, 12, 16xpcbas 15087 . . . . 5  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
23 eqid 2451 . . . . 5  |-  ( Hom  `  ( C  X.c  D ) )  =  ( Hom  `  ( C  X.c  D ) )
24 opelxpi 4966 . . . . . 6  |-  ( ( X  e.  A  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
2517, 19, 24syl2anc 661 . . . . 5  |-  ( ph  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
2613, 22, 23, 3, 4, 14, 251stf1 15101 . . . 4  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )  =  ( 1st `  <. X ,  Y >. )
)
2721, 26syl5eq 2503 . . 3  |-  ( ph  ->  ( X ( 1st `  ( C  1stF  D )
) Y )  =  ( 1st `  <. X ,  Y >. )
)
28 op1stg 6686 . . . 4  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2917, 19, 28syl2anc 661 . . 3  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
3027, 29eqtrd 2491 . 2  |-  ( ph  ->  ( X ( 1st `  ( C  1stF  D )
) Y )  =  X )
3110, 20, 303eqtrd 2495 1  |-  ( ph  ->  ( ( 1st `  K
) `  Y )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   <.cop 3978    X. cxp 4933   ` cfv 5513  (class class class)co 6187   1stc1st 6672   Basecbs 14273   Hom chom 14348   Catccat 14701    X.c cxpc 15077    1stF c1stf 15078   curryF ccurf 15119  Δfunccdiag 15121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-map 7313  df-ixp 7361  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-4 10480  df-5 10481  df-6 10482  df-7 10483  df-8 10484  df-9 10485  df-10 10486  df-n0 10678  df-z 10745  df-dec 10854  df-uz 10960  df-fz 11536  df-struct 14275  df-ndx 14276  df-slot 14277  df-base 14278  df-hom 14361  df-cco 14362  df-cat 14705  df-cid 14706  df-func 14867  df-xpc 15081  df-1stf 15082  df-curf 15123  df-diag 15125
This theorem is referenced by:  curf2ndf  15156
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