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Theorem cmp2morpgrp 25963
Description: Graph of the 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
cmp2morpgrp  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.graph  `  ( A O B ) )  =  ( (.graph  `  A
)  o.  (.graph  `  B ) ) )

Proof of Theorem cmp2morpgrp
StepHypRef Expression
1 cmp2morpcatt.1 . . . 4  |-  O  =  ( ro SetCat `  U
)
2 cmp2morpcatt.2 . . . 4  |- .Morphism  =  ( Morphism SetCat `  U )
3 cmp2morpcatt.3 . . . 4  |- .dom  =  ( dom SetCat `  U
)
4 cmp2morpcatt.4 . . . 4  |- .cod  =  ( cod SetCat `  U
)
5 eqid 2283 . . . 4  |-  ( graph SetCat `  U )  =  (
graph SetCat `  U )
61, 2, 3, 4, 5cmp2morpcatt 25962 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( ( graph SetCat `  U
) `  A )  o.  ( ( graph SetCat `  U
) `  B )
) >. )
76fveq2d 5529 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.graph  `  ( A O B ) )  =  (.graph  `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (
( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) ) >.
) )
8 simp1 955 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  U  e.  Univ )
91, 2, 3, 4rocatval2 25960 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e. .Morphism  )
106, 9eqeltrrd 2358 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (
( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) ) >.  e. .Morphism  )
11 cmp2morpcatt.5 . . . 4  |- .graph  =  ( graph SetCat `  U )
1211, 2isgraphmrph2 25924 . . 3  |-  ( ( U  e.  Univ  /\  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( ( graph SetCat `  U
) `  A )  o.  ( ( graph SetCat `  U
) `  B )
) >.  e. .Morphism  )  ->  (.graph  `  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( ( graph SetCat `  U
) `  A )  o.  ( ( graph SetCat `  U
) `  B )
) >. )  =  ( 2nd `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (
( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) ) >.
) )
138, 10, 12syl2anc 642 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.graph  ` 
<. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (
( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) ) >.
)  =  ( 2nd `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (
( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) ) >.
) )
14 opex 4237 . . . . 5  |-  <. (.dom  `  B ) ,  (.cod  `  A )
>.  e.  _V
15 fvex 5539 . . . . . 6  |-  ( (
graph SetCat `  U ) `  A )  e.  _V
16 fvex 5539 . . . . . 6  |-  ( (
graph SetCat `  U ) `  B )  e.  _V
1715, 16coex 5216 . . . . 5  |-  ( ( ( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) )  e. 
_V
1814, 17op2nd 6129 . . . 4  |-  ( 2nd `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (
( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) ) >.
)  =  ( ( ( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) )
1911fveq1i 5526 . . . . . 6  |-  (.graph  `  A )  =  ( ( graph SetCat `  U ) `  A )
2019eqcomi 2287 . . . . 5  |-  ( (
graph SetCat `  U ) `  A )  =  (.graph  `  A )
2111fveq1i 5526 . . . . . 6  |-  (.graph  `  B )  =  ( ( graph SetCat `  U ) `  B )
2221eqcomi 2287 . . . . 5  |-  ( (
graph SetCat `  U ) `  B )  =  (.graph  `  B )
2320, 22coeq12i 4847 . . . 4  |-  ( ( ( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) )  =  ( (.graph  `  A )  o.  (.graph  `  B ) )
2418, 23eqtri 2303 . . 3  |-  ( 2nd `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (
( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) ) >.
)  =  ( (.graph  `  A )  o.  (.graph  `  B ) )
2524a1i 10 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( 2nd ` 
<. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( (
( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) ) >.
)  =  ( (.graph  `  A )  o.  (.graph  `  B ) ) )
267, 13, 253eqtrd 2319 1  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.graph  `  ( A O B ) )  =  ( (.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:  cmpmorass  25966
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-map 6774  df-morcatset 25911  df-domcatset 25920  df-graphcatset 25922  df-codcatset 25933  df-rocatset 25956
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