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Theorem arwrid 13921
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 2296 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 arwlid.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X H Y ) )
4 arwlid.h . . . . . . . 8  |-  H  =  (Homa
`  C )
54homarcl 13876 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
63, 5syl 15 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
7 eqid 2296 . . . . . 6  |-  ( Id
`  C )  =  ( Id `  C
)
84, 2homarcl2 13883 . . . . . . . 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 13907 . . . . 5  |-  ( ph  ->  ( 2nd `  (  .1.  `  X ) )  =  ( ( Id
`  C ) `  X ) )
1211oveq2d 5890 . . . 4  |-  ( ph  ->  ( ( 2nd `  F
) ( <. X ,  X >. (comp `  C
) Y ) ( 2nd `  (  .1.  `  X ) ) )  =  ( ( 2nd `  F ) ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) ) )
13 eqid 2296 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
14 eqid 2296 . . . . 5  |-  (comp `  C )  =  (comp `  C )
159simprd 449 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
164, 13homahom 13887 . . . . . 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 13602 . . . 4  |-  ( ph  ->  ( ( 2nd `  F
) ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
)  =  ( 2nd `  F ) )
1912, 18eqtrd 2328 . . 3  |-  ( ph  ->  ( ( 2nd `  F
) ( <. X ,  X >. (comp `  C
) Y ) ( 2nd `  (  .1.  `  X ) ) )  =  ( 2nd `  F
) )
2019oteq3d 3826 . 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 13908 . . 3  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X H X ) )
2321, 4, 22, 3, 14coaval 13916 . 2  |-  ( ph  ->  ( F  .x.  (  .1.  `  X ) )  =  <. X ,  Y ,  ( ( 2nd `  F ) ( <. X ,  X >. (comp `  C ) Y ) ( 2nd `  (  .1.  `  X ) ) ) >. )
244homadmcd 13890 . . 3  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )
253, 24syl 15 . 2  |-  ( ph  ->  F  =  <. X ,  Y ,  ( 2nd `  F ) >. )
2620, 23, 253eqtr4d 2338 1  |-  ( ph  ->  ( F  .x.  (  .1.  `  X ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656   <.cotp 3657   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583  Homachoma 13871  Idacida 13901  compaccoa 13902
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-cat 13586  df-cid 13587  df-doma 13872  df-coda 13873  df-homa 13874  df-arw 13875  df-ida 13903  df-coa 13904
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