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Theorem arwrid 13905
Description: Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwlid.h  |-  H  =  (Homa
`  C )
arwlid.o  |-  .x.  =  (compa `  C )
arwlid.a  |-  .1.  =  (Ida `  C )
arwlid.f  |-  ( ph  ->  F  e.  ( X H Y ) )
Assertion
Ref Expression
arwrid  |-  ( ph  ->  ( F  .x.  (  .1.  `  X ) )  =  F )

Proof of Theorem arwrid
StepHypRef Expression
1 arwlid.a . . . . . 6  |-  .1.  =  (Ida `  C )
2 eqid 2283 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 arwlid.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X H Y ) )
4 arwlid.h . . . . . . . 8  |-  H  =  (Homa
`  C )
54homarcl 13860 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
63, 5syl 15 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
7 eqid 2283 . . . . . 6  |-  ( Id
`  C )  =  ( Id `  C
)
84, 2homarcl2 13867 . . . . . . . 8  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
93, 8syl 15 . . . . . . 7  |-  ( ph  ->  ( X  e.  (
Base `  C )  /\  Y  e.  ( Base `  C ) ) )
109simpld 445 . . . . . 6  |-  ( ph  ->  X  e.  ( Base `  C ) )
111, 2, 6, 7, 10ida2 13891 . . . . 5  |-  ( ph  ->  ( 2nd `  (  .1.  `  X ) )  =  ( ( Id
`  C ) `  X ) )
1211oveq2d 5874 . . . 4  |-  ( ph  ->  ( ( 2nd `  F
) ( <. X ,  X >. (comp `  C
) Y ) ( 2nd `  (  .1.  `  X ) ) )  =  ( ( 2nd `  F ) ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) ) )
13 eqid 2283 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
14 eqid 2283 . . . . 5  |-  (comp `  C )  =  (comp `  C )
159simprd 449 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
164, 13homahom 13871 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X (  Hom  `  C ) Y ) )
173, 16syl 15 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  e.  ( X (  Hom  `  C
) Y ) )
182, 13, 7, 6, 10, 14, 15, 17catrid 13586 . . . 4  |-  ( ph  ->  ( ( 2nd `  F
) ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
)  =  ( 2nd `  F ) )
1912, 18eqtrd 2315 . . 3  |-  ( ph  ->  ( ( 2nd `  F
) ( <. X ,  X >. (comp `  C
) Y ) ( 2nd `  (  .1.  `  X ) ) )  =  ( 2nd `  F
) )
2019oteq3d 3810 . 2  |-  ( ph  -> 
<. X ,  Y , 
( ( 2nd `  F
) ( <. X ,  X >. (comp `  C
) Y ) ( 2nd `  (  .1.  `  X ) ) )
>.  =  <. X ,  Y ,  ( 2nd `  F ) >. )
21 arwlid.o . . 3  |-  .x.  =  (compa `  C )
221, 2, 6, 10, 4idahom 13892 . . 3  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X H X ) )
2321, 4, 22, 3, 14coaval 13900 . 2  |-  ( ph  ->  ( F  .x.  (  .1.  `  X ) )  =  <. X ,  Y ,  ( ( 2nd `  F ) ( <. X ,  X >. (comp `  C ) Y ) ( 2nd `  (  .1.  `  X ) ) ) >. )
244homadmcd 13874 . . 3  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )
253, 24syl 15 . 2  |-  ( ph  ->  F  =  <. X ,  Y ,  ( 2nd `  F ) >. )
2620, 23, 253eqtr4d 2325 1  |-  ( ph  ->  ( F  .x.  (  .1.  `  X ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   <.cotp 3644   ` cfv 5255  (class class class)co 5858   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567  Homachoma 13855  Idacida 13885  compaccoa 13886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-cat 13570  df-cid 13571  df-doma 13856  df-coda 13857  df-homa 13858  df-arw 13859  df-ida 13887  df-coa 13888
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