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Theorem cmp2morpcatt 25962
Description: Composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
Hypotheses
Ref Expression
cmp2morpcatt.1  |-  O  =  ( ro SetCat `  U
)
cmp2morpcatt.2  |- .Morphism  =  ( Morphism SetCat `  U )
cmp2morpcatt.3  |- .dom  =  ( dom SetCat `  U
)
cmp2morpcatt.4  |- .cod  =  ( cod SetCat `  U
)
cmp2morpcatt.5  |- .graph  =  ( graph SetCat `  U )
Assertion
Ref Expression
cmp2morpcatt  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( (.graph  `  A
)  o.  (.graph  `  B ) ) >.
)

Proof of Theorem cmp2morpcatt
StepHypRef Expression
1 cmp2morpcatt.5 . . . . 5  |- .graph  =  ( graph SetCat `  U )
2 cmp2morpcatt.2 . . . . 5  |- .Morphism  =  ( Morphism SetCat `  U )
31, 2isgraphmrph2 25924 . . . 4  |-  ( ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (.graph  `  A )  =  ( 2nd `  A ) )
41, 2isgraphmrph2 25924 . . . 4  |-  ( ( U  e.  Univ  /\  B  e. .Morphism  )  ->  (.graph  `  B )  =  ( 2nd `  B ) )
53, 4anim12dan 810 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  ) )  -> 
( (.graph  `  A )  =  ( 2nd `  A )  /\  (.graph  `  B )  =  ( 2nd `  B ) ) )
653adant3 975 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( (.graph  `  A )  =  ( 2nd `  A )  /\  (.graph  `  B )  =  ( 2nd `  B ) ) )
7 cmp2morpcatt.1 . . . 4  |-  O  =  ( ro SetCat `  U
)
8 cmp2morpcatt.3 . . . 4  |- .dom  =  ( dom SetCat `  U
)
9 cmp2morpcatt.4 . . . 4  |- .cod  =  ( cod SetCat `  U
)
107, 2, 8, 9cmp2morpcats 25961 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
)
11 simpl 443 . . . . . 6  |-  ( ( (.graph  `  A
)  =  ( 2nd `  A )  /\  (.graph  `  B )  =  ( 2nd `  B ) )  ->  (.graph  `  A )  =  ( 2nd `  A ) )
12 simpr 447 . . . . . 6  |-  ( ( (.graph  `  A
)  =  ( 2nd `  A )  /\  (.graph  `  B )  =  ( 2nd `  B ) )  ->  (.graph  `  B )  =  ( 2nd `  B ) )
1311, 12coeq12d 4848 . . . . 5  |-  ( ( (.graph  `  A
)  =  ( 2nd `  A )  /\  (.graph  `  B )  =  ( 2nd `  B ) )  ->  ( (.graph  `  A )  o.  (.graph  `  B ) )  =  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) )
1413opeq2d 3803 . . . 4  |-  ( ( (.graph  `  A
)  =  ( 2nd `  A )  /\  (.graph  `  B )  =  ( 2nd `  B ) )  ->  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (.graph  `  A )  o.  (.graph  `  B ) ) >.  =  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
1514eqeq2d 2294 . . 3  |-  ( ( (.graph  `  A
)  =  ( 2nd `  A )  /\  (.graph  `  B )  =  ( 2nd `  B ) )  ->  ( ( A O B )  = 
<. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (.graph  `  A )  o.  (.graph  `  B ) ) >.  <->  ( A O B )  =  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
)
1610, 15syl5ibr 212 . 2  |-  ( ( (.graph  `  A
)  =  ( 2nd `  A )  /\  (.graph  `  B )  =  ( 2nd `  B ) )  ->  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( (.graph  `  A
)  o.  (.graph  `  B ) ) >.
) )
176, 16mpcom 32 1  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( (.graph  `  A
)  o.  (.graph  `  B ) ) >.
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643    o. ccom 4693   ` cfv 5255  (class class class)co 5858   2ndc2nd 6121   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   graph SetCatcgraphcase 25921   cod
SetCatccodcase 25932   ro SetCatcrocase 25955
This theorem is referenced by:  cmp2morpgrp  25963  cmpmorass  25966  cmpidmor3  25970
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-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-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-domcatset 25920  df-graphcatset 25922  df-codcatset 25933  df-rocatset 25956
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