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Theorem cmpmorass 25966
Description: Associativity of composition in category Set. (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
)
Assertion
Ref Expression
cmpmorass  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  ( C O ( B O A ) )  =  ( ( C O B ) O A ) )

Proof of Theorem cmpmorass
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  U  e.  Univ )
2 simp23 990 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  C  e. .Morphism  )
3 simp22 989 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  B  e. .Morphism  )
4 simp21 988 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  A  e. .Morphism  )
5 simp3r 984 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (.dom  `  B )  =  (.cod  `  A ) )
6 cmp2morpcatt.1 . . . . 5  |-  O  =  ( ro SetCat `  U
)
7 cmp2morpcatt.2 . . . . 5  |- .Morphism  =  ( Morphism SetCat `  U )
8 cmp2morpcatt.3 . . . . 5  |- .dom  =  ( dom SetCat `  U
)
9 cmp2morpcatt.4 . . . . 5  |- .cod  =  ( cod SetCat `  U
)
106, 7, 8, 9rocatval2 25960 . . . 4  |-  ( ( U  e.  Univ  /\  ( B  e. .Morphism  /\  A  e. .Morphism  )  /\  (.dom  `  B )  =  (.cod  `  A ) )  ->  ( B O A )  e. .Morphism  )
111, 3, 4, 5, 10syl121anc 1187 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  ( B O A )  e. .Morphism  )
12 simp3l 983 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (.dom  `  C )  =  (.cod  `  B ) )
136, 7, 8, 9cmp2morpcod 25965 . . . . 5  |-  ( ( U  e.  Univ  /\  ( B  e. .Morphism  /\  A  e. .Morphism  )  /\  (.dom  `  B )  =  (.cod  `  A ) )  ->  (.cod  `  ( B O A ) )  =  (.cod  `  B ) )
141, 3, 4, 5, 13syl121anc 1187 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (.cod  `  ( B O A ) )  =  (.cod  `  B ) )
1512, 14eqtr4d 2318 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (.dom  `  C )  =  (.cod  `  ( B O A ) ) )
16 eqid 2283 . . . 4  |-  ( graph SetCat `  U )  =  (
graph SetCat `  U )
176, 7, 8, 9, 16cmp2morpcatt 25962 . . 3  |-  ( ( U  e.  Univ  /\  ( C  e. .Morphism  /\  ( B O A )  e. .Morphism  )  /\  (.dom  `  C )  =  (.cod  `  ( B O A ) ) )  ->  ( C O ( B O A ) )  = 
<. <. (.dom  `  ( B O A ) ) ,  (.cod  `  C ) >. ,  ( ( ( graph SetCat `  U
) `  C )  o.  ( ( graph SetCat `  U
) `  ( B O A ) ) )
>. )
181, 2, 11, 15, 17syl121anc 1187 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  ( C O ( B O A ) )  = 
<. <. (.dom  `  ( B O A ) ) ,  (.cod  `  C ) >. ,  ( ( ( graph SetCat `  U
) `  C )  o.  ( ( graph SetCat `  U
) `  ( B O A ) ) )
>. )
196, 7, 8, 9cmp2morpdom 25964 . . . . 5  |-  ( ( U  e.  Univ  /\  ( B  e. .Morphism  /\  A  e. .Morphism  )  /\  (.dom  `  B )  =  (.cod  `  A ) )  ->  (.dom  `  ( B O A ) )  =  (.dom  `  A ) )
201, 3, 4, 5, 19syl121anc 1187 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (.dom  `  ( B O A ) )  =  (.dom  `  A ) )
2120opeq1d 3802 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  <. (.dom  `  ( B O A ) ) ,  (.cod  `  C )
>.  =  <. (.dom  `  A ) ,  (.cod  `  C )
>. )
226, 7, 8, 9, 16cmp2morpgrp 25963 . . . . 5  |-  ( ( U  e.  Univ  /\  ( B  e. .Morphism  /\  A  e. .Morphism  )  /\  (.dom  `  B )  =  (.cod  `  A ) )  ->  ( ( graph
SetCat `  U ) `  ( B O A ) )  =  ( ( ( graph SetCat `  U ) `  B )  o.  (
( graph SetCat `  U ) `  A ) ) )
231, 3, 4, 5, 22syl121anc 1187 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (
( graph SetCat `  U ) `  ( B O A ) )  =  ( ( ( graph SetCat `  U
) `  B )  o.  ( ( graph SetCat `  U
) `  A )
) )
2423coeq2d 4846 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (
( ( graph SetCat `  U
) `  C )  o.  ( ( graph SetCat `  U
) `  ( B O A ) ) )  =  ( ( (
graph SetCat `  U ) `  C )  o.  (
( ( graph SetCat `  U
) `  B )  o.  ( ( graph SetCat `  U
) `  A )
) ) )
2521, 24opeq12d 3804 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  <. <. (.dom  `  ( B O A ) ) ,  (.cod  `  C )
>. ,  ( (
( graph SetCat `  U ) `  C )  o.  (
( graph SetCat `  U ) `  ( B O A ) ) ) >.  =  <. <. (.dom  `  A ) ,  (.cod  `  C )
>. ,  ( (
( graph SetCat `  U ) `  C )  o.  (
( ( graph SetCat `  U
) `  B )  o.  ( ( graph SetCat `  U
) `  A )
) ) >. )
266, 7, 8, 9cmp2morpcod 25965 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( C  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  C )  =  (.cod  `  B ) )  ->  (.cod  `  ( C O B ) )  =  (.cod  `  C ) )
2726eqcomd 2288 . . . . . 6  |-  ( ( U  e.  Univ  /\  ( C  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  C )  =  (.cod  `  B ) )  ->  (.cod  `  C )  =  (.cod  `  ( C O B ) ) )
281, 2, 3, 12, 27syl121anc 1187 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (.cod  `  C )  =  (.cod  `  ( C O B ) ) )
2928opeq2d 3803 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  <. (.dom  `  A ) ,  (.cod  `  C )
>.  =  <. (.dom  `  A ) ,  (.cod  `  ( C O B ) )
>. )
306, 7, 8, 9, 16cmp2morpgrp 25963 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( C  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  C )  =  (.cod  `  B ) )  ->  ( ( graph
SetCat `  U ) `  ( C O B ) )  =  ( ( ( graph SetCat `  U ) `  C )  o.  (
( graph SetCat `  U ) `  B ) ) )
311, 2, 3, 12, 30syl121anc 1187 . . . . . 6  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (
( graph SetCat `  U ) `  ( C O B ) )  =  ( ( ( graph SetCat `  U
) `  C )  o.  ( ( graph SetCat `  U
) `  B )
) )
3231eqcomd 2288 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (
( ( graph SetCat `  U
) `  C )  o.  ( ( graph SetCat `  U
) `  B )
)  =  ( (
graph SetCat `  U ) `  ( C O B ) ) )
3332coeq1d 4845 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (
( ( ( graph SetCat `  U ) `  C
)  o.  ( (
graph SetCat `  U ) `  B ) )  o.  ( ( graph SetCat `  U
) `  A )
)  =  ( ( ( graph SetCat `  U ) `  ( C O B ) )  o.  (
( graph SetCat `  U ) `  A ) ) )
3429, 33opeq12d 3804 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  <. <. (.dom  `  A ) ,  (.cod  `  C )
>. ,  ( (
( ( graph SetCat `  U
) `  C )  o.  ( ( graph SetCat `  U
) `  B )
)  o.  ( (
graph SetCat `  U ) `  A ) ) >.  =  <. <. (.dom  `  A ) ,  (.cod  `  ( C O B ) )
>. ,  ( (
( graph SetCat `  U ) `  ( C O B ) )  o.  (
( graph SetCat `  U ) `  A ) ) >.
)
35 coass 5191 . . . . . 6  |-  ( ( ( ( graph SetCat `  U
) `  C )  o.  ( ( graph SetCat `  U
) `  B )
)  o.  ( (
graph SetCat `  U ) `  A ) )  =  ( ( ( graph SetCat `  U ) `  C
)  o.  ( ( ( graph SetCat `  U ) `  B )  o.  (
( graph SetCat `  U ) `  A ) ) )
3635eqcomi 2287 . . . . 5  |-  ( ( ( graph SetCat `  U ) `  C )  o.  (
( ( graph SetCat `  U
) `  B )  o.  ( ( graph SetCat `  U
) `  A )
) )  =  ( ( ( ( graph SetCat `  U ) `  C
)  o.  ( (
graph SetCat `  U ) `  B ) )  o.  ( ( graph SetCat `  U
) `  A )
)
3736a1i 10 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (
( ( graph SetCat `  U
) `  C )  o.  ( ( ( graph SetCat `  U ) `  B
)  o.  ( (
graph SetCat `  U ) `  A ) ) )  =  ( ( ( ( graph SetCat `  U ) `  C )  o.  (
( graph SetCat `  U ) `  B ) )  o.  ( ( graph SetCat `  U
) `  A )
) )
3837opeq2d 3803 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  <. <. (.dom  `  A ) ,  (.cod  `  C )
>. ,  ( (
( graph SetCat `  U ) `  C )  o.  (
( ( graph SetCat `  U
) `  B )  o.  ( ( graph SetCat `  U
) `  A )
) ) >.  =  <. <.
(.dom  `  A
) ,  (.cod  `  C ) >. ,  ( ( ( ( graph SetCat `  U ) `  C
)  o.  ( (
graph SetCat `  U ) `  B ) )  o.  ( ( graph SetCat `  U
) `  A )
) >. )
396, 7, 8, 9rocatval2 25960 . . . . 5  |-  ( ( U  e.  Univ  /\  ( C  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  C )  =  (.cod  `  B ) )  ->  ( C O B )  e. .Morphism  )
401, 2, 3, 12, 39syl121anc 1187 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  ( C O B )  e. .Morphism  )
416, 7, 8, 9cmp2morpdom 25964 . . . . . 6  |-  ( ( U  e.  Univ  /\  ( C  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  C )  =  (.cod  `  B ) )  ->  (.dom  `  ( C O B ) )  =  (.dom  `  B ) )
421, 2, 3, 12, 41syl121anc 1187 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (.dom  `  ( C O B ) )  =  (.dom  `  B ) )
4342, 5eqtrd 2315 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (.dom  `  ( C O B ) )  =  (.cod  `  A ) )
446, 7, 8, 9, 16cmp2morpcatt 25962 . . . 4  |-  ( ( U  e.  Univ  /\  (
( C O B )  e. .Morphism  /\  A  e. .Morphism  )  /\  (.dom  `  ( C O B ) )  =  (.cod  `  A ) )  -> 
( ( C O B ) O A )  =  <. <. (.dom  `  A ) ,  (.cod  `  ( C O B ) )
>. ,  ( (
( graph SetCat `  U ) `  ( C O B ) )  o.  (
( graph SetCat `  U ) `  A ) ) >.
)
451, 40, 4, 43, 44syl121anc 1187 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  (
( C O B ) O A )  =  <. <. (.dom  `  A ) ,  (.cod  `  ( C O B ) )
>. ,  ( (
( graph SetCat `  U ) `  ( C O B ) )  o.  (
( graph SetCat `  U ) `  A ) ) >.
)
4634, 38, 453eqtr4d 2325 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  <. <. (.dom  `  A ) ,  (.cod  `  C )
>. ,  ( (
( graph SetCat `  U ) `  C )  o.  (
( ( graph SetCat `  U
) `  B )  o.  ( ( graph SetCat `  U
) `  A )
) ) >.  =  ( ( C O B ) O A ) )
4718, 25, 463eqtrd 2319 1  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B )  /\  (.dom  `  B )  =  (.cod  `  A ) ) )  ->  ( C O ( B O A ) )  =  ( ( C O B ) O A ) )
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   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   graph SetCatcgraphcase 25921   cod
SetCatccodcase 25932   ro SetCatcrocase 25955
This theorem is referenced by:  setiscat  25979
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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