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Theorem fucrid 15185
Description: Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fuclid.q  |-  Q  =  ( C FuncCat  D )
fuclid.n  |-  N  =  ( C Nat  D )
fuclid.x  |-  .xb  =  (comp `  Q )
fuclid.1  |-  .1.  =  ( Id `  D )
fuclid.r  |-  ( ph  ->  R  e.  ( F N G ) )
Assertion
Ref Expression
fucrid  |-  ( ph  ->  ( R ( <. F ,  F >.  .xb 
G ) (  .1. 
o.  ( 1st `  F
) ) )  =  R )

Proof of Theorem fucrid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2462 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2462 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
3 relfunc 15080 . . . . . . . 8  |-  Rel  ( C  Func  D )
4 fuclid.r . . . . . . . . . 10  |-  ( ph  ->  R  e.  ( F N G ) )
5 fuclid.n . . . . . . . . . . 11  |-  N  =  ( C Nat  D )
65natrcl 15168 . . . . . . . . . 10  |-  ( R  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
74, 6syl 16 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
87simpld 459 . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
9 1st2ndbr 6825 . . . . . . . 8  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
103, 8, 9sylancr 663 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
111, 2, 10funcf1 15084 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
12 fvco3 5937 . . . . . 6  |-  ( ( ( 1st `  F
) : ( Base `  C ) --> ( Base `  D )  /\  x  e.  ( Base `  C
) )  ->  (
(  .1.  o.  ( 1st `  F ) ) `
 x )  =  (  .1.  `  (
( 1st `  F
) `  x )
) )
1311, 12sylan 471 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (  .1.  o.  ( 1st `  F
) ) `  x
)  =  (  .1.  `  ( ( 1st `  F
) `  x )
) )
1413oveq2d 6293 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( R `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 x ) >.
(comp `  D )
( ( 1st `  G
) `  x )
) ( (  .1. 
o.  ( 1st `  F
) ) `  x
) )  =  ( ( R `  x
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) (  .1.  `  ( ( 1st `  F
) `  x )
) ) )
15 eqid 2462 . . . . 5  |-  ( Hom  `  D )  =  ( Hom  `  D )
16 fuclid.1 . . . . 5  |-  .1.  =  ( Id `  D )
17 funcrcl 15081 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
188, 17syl 16 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1918simprd 463 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
2019adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
2111ffvelrnda 6014 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
22 eqid 2462 . . . . 5  |-  (comp `  D )  =  (comp `  D )
237simprd 463 . . . . . . . 8  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
24 1st2ndbr 6825 . . . . . . . 8  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
253, 23, 24sylancr 663 . . . . . . 7  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
261, 2, 25funcf1 15084 . . . . . 6  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
2726ffvelrnda 6014 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  D )
)
285, 4nat1st2nd 15169 . . . . . . 7  |-  ( ph  ->  R  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
2928adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  R  e.  ( <. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
30 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
315, 29, 1, 15, 30natcl 15171 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( R `  x )  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  G
) `  x )
) )
322, 15, 16, 20, 21, 22, 27, 31catrid 14930 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( R `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 x ) >.
(comp `  D )
( ( 1st `  G
) `  x )
) (  .1.  `  ( ( 1st `  F
) `  x )
) )  =  ( R `  x ) )
3314, 32eqtrd 2503 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( R `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 x ) >.
(comp `  D )
( ( 1st `  G
) `  x )
) ( (  .1. 
o.  ( 1st `  F
) ) `  x
) )  =  ( R `  x ) )
3433mpteq2dva 4528 . 2  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  ( ( R `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( (  .1.  o.  ( 1st `  F ) ) `  x ) ) )  =  ( x  e.  ( Base `  C
)  |->  ( R `  x ) ) )
35 fuclid.q . . 3  |-  Q  =  ( C FuncCat  D )
36 fuclid.x . . 3  |-  .xb  =  (comp `  Q )
3735, 5, 16, 8fucidcl 15183 . . 3  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  ( F N F ) )
3835, 5, 1, 22, 36, 37, 4fucco 15180 . 2  |-  ( ph  ->  ( R ( <. F ,  F >.  .xb 
G ) (  .1. 
o.  ( 1st `  F
) ) )  =  ( x  e.  (
Base `  C )  |->  ( ( R `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( (  .1.  o.  ( 1st `  F ) ) `  x ) ) ) )
395, 28, 1natfn 15172 . . 3  |-  ( ph  ->  R  Fn  ( Base `  C ) )
40 dffn5 5906 . . 3  |-  ( R  Fn  ( Base `  C
)  <->  R  =  (
x  e.  ( Base `  C )  |->  ( R `
 x ) ) )
4139, 40sylib 196 . 2  |-  ( ph  ->  R  =  ( x  e.  ( Base `  C
)  |->  ( R `  x ) ) )
4234, 38, 413eqtr4d 2513 1  |-  ( ph  ->  ( R ( <. F ,  F >.  .xb 
G ) (  .1. 
o.  ( 1st `  F
) ) )  =  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   <.cop 4028   class class class wbr 4442    |-> cmpt 4500    o. ccom 4998   Rel wrel 4999    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   1stc1st 6774   2ndc2nd 6775   Basecbs 14481   Hom chom 14557  compcco 14558   Catccat 14910   Idccid 14911    Func cfunc 15072   Nat cnat 15159   FuncCat cfuc 15160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-hom 14570  df-cco 14571  df-cat 14914  df-cid 14915  df-func 15076  df-nat 15161  df-fuc 15162
This theorem is referenced by:  fuccatid  15187
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