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Theorem fuclid 15382
Description: Left 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
fuclid  |-  ( ph  ->  ( (  .1.  o.  ( 1st `  G ) ) ( <. F ,  G >.  .xb  G ) R )  =  R )

Proof of Theorem fuclid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2457 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
3 relfunc 15278 . . . . . . . 8  |-  Rel  ( C  Func  D )
4 fuclid.r . . . . . . . . . 10  |-  ( ph  ->  R  e.  ( F N G ) )
5 fuclid.n . . . . . . . . . . 11  |-  N  =  ( C Nat  D )
65natrcl 15366 . . . . . . . . . 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 ) ) )
87simprd 463 . . . . . . . 8  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
9 1st2ndbr 6848 . . . . . . . 8  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
103, 8, 9sylancr 663 . . . . . . 7  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
111, 2, 10funcf1 15282 . . . . . 6  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
12 fvco3 5950 . . . . . 6  |-  ( ( ( 1st `  G
) : ( Base `  C ) --> ( Base `  D )  /\  x  e.  ( Base `  C
) )  ->  (
(  .1.  o.  ( 1st `  G ) ) `
 x )  =  (  .1.  `  (
( 1st `  G
) `  x )
) )
1311, 12sylan 471 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (  .1.  o.  ( 1st `  G
) ) `  x
)  =  (  .1.  `  ( ( 1st `  G
) `  x )
) )
1413oveq1d 6311 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
(  .1.  o.  ( 1st `  G ) ) `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( R `
 x ) )  =  ( (  .1.  `  ( ( 1st `  G
) `  x )
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( R `
 x ) ) )
15 eqid 2457 . . . . 5  |-  ( Hom  `  D )  =  ( Hom  `  D )
16 fuclid.1 . . . . 5  |-  .1.  =  ( Id `  D )
177simpld 459 . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
18 funcrcl 15279 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1917, 18syl 16 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
2019simprd 463 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
2120adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
22 1st2ndbr 6848 . . . . . . . 8  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
233, 17, 22sylancr 663 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
241, 2, 23funcf1 15282 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2524ffvelrnda 6032 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
26 eqid 2457 . . . . 5  |-  (comp `  D )  =  (comp `  D )
2711ffvelrnda 6032 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  D )
)
285, 4nat1st2nd 15367 . . . . . . 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 15369 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( R `  x )  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  G
) `  x )
) )
322, 15, 16, 21, 25, 26, 27, 31catlid 15100 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (  .1.  `  ( ( 1st `  G ) `  x
) ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( R `
 x ) )  =  ( R `  x ) )
3314, 32eqtrd 2498 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
(  .1.  o.  ( 1st `  G ) ) `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( R `
 x ) )  =  ( R `  x ) )
3433mpteq2dva 4543 . 2  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  ( ( (  .1. 
o.  ( 1st `  G
) ) `  x
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( R `
 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 15381 . . 3  |-  ( ph  ->  (  .1.  o.  ( 1st `  G ) )  e.  ( G N G ) )
3835, 5, 1, 26, 36, 4, 37fucco 15378 . 2  |-  ( ph  ->  ( (  .1.  o.  ( 1st `  G ) ) ( <. F ,  G >.  .xb  G ) R )  =  ( x  e.  ( Base `  C
)  |->  ( ( (  .1.  o.  ( 1st `  G ) ) `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  x
) ) ( R `
 x ) ) ) )
395, 28, 1natfn 15370 . . 3  |-  ( ph  ->  R  Fn  ( Base `  C ) )
40 dffn5 5918 . . 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 2508 1  |-  ( ph  ->  ( (  .1.  o.  ( 1st `  G ) ) ( <. F ,  G >.  .xb  G ) R )  =  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    o. ccom 5012   Rel wrel 5013    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   Basecbs 14644   Hom chom 14723  compcco 14724   Catccat 15081   Idccid 15082    Func cfunc 15270   Nat cnat 15357   FuncCat cfuc 15358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-hom 14736  df-cco 14737  df-cat 15085  df-cid 15086  df-func 15274  df-nat 15359  df-fuc 15360
This theorem is referenced by:  fuccatid  15385
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