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Theorem cmp2morpgrp 26066
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 2296 . . . 4  |-  ( graph SetCat `  U )  =  (
graph SetCat `  U )
61, 2, 3, 4, 5cmp2morpcatt 26065 . . 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 5545 . 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 26063 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e. .Morphism  )
106, 9eqeltrrd 2371 . . 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 26027 . . 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 4253 . . . . 5  |-  <. (.dom  `  B ) ,  (.cod  `  A )
>.  e.  _V
15 fvex 5555 . . . . . 6  |-  ( (
graph SetCat `  U ) `  A )  e.  _V
16 fvex 5555 . . . . . 6  |-  ( (
graph SetCat `  U ) `  B )  e.  _V
1715, 16coex 5232 . . . . 5  |-  ( ( ( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) )  e. 
_V
1814, 17op2nd 6145 . . . 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 5542 . . . . . 6  |-  (.graph  `  A )  =  ( ( graph SetCat `  U ) `  A )
2019eqcomi 2300 . . . . 5  |-  ( (
graph SetCat `  U ) `  A )  =  (.graph  `  A )
2111fveq1i 5542 . . . . . 6  |-  (.graph  `  B )  =  ( ( graph SetCat `  U ) `  B )
2221eqcomi 2300 . . . . 5  |-  ( (
graph SetCat `  U ) `  B )  =  (.graph  `  B )
2320, 22coeq12i 4863 . . . 4  |-  ( ( ( graph SetCat `  U ) `  A )  o.  (
( graph SetCat `  U ) `  B ) )  =  ( (.graph  `  A )  o.  (.graph  `  B ) )
2418, 23eqtri 2316 . . 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 2332 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 1632    e. wcel 1696   <.cop 3656    o. ccom 4709   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137   Univcgru 8428   Morphism SetCatccmrcase 26013   dom
SetCatcdomcase 26022   graph SetCatcgraphcase 26024   cod
SetCatccodcase 26035   ro SetCatcrocase 26058
This theorem is referenced by:  cmpmorass  26069
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-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-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-map 6790  df-morcatset 26014  df-domcatset 26023  df-graphcatset 26025  df-codcatset 26036  df-rocatset 26059
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