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Theorem cmpidmor3 25970
Description: Composition with an identity. (Contributed by FL, 8-Nov-2013.)
Hypotheses
Ref Expression
cmpidmor3.1  |-  O  =  ( ro SetCat `  U
)
cmpidmor3.2  |- .Morphism  =  ( Morphism SetCat `  U )
cmpidmor3.4  |- .dom  =  ( dom SetCat `  U
)
cmpidmor3.5  |- .id  =  ( Id SetCat `  U
)
Assertion
Ref Expression
cmpidmor3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( F O (.id  `  (.dom  `  F
) ) )  =  F )

Proof of Theorem cmpidmor3
StepHypRef Expression
1 cmpidmor3.2 . . . . . 6  |- .Morphism  =  ( Morphism SetCat `  U )
2 cmpidmor3.4 . . . . . 6  |- .dom  =  ( dom SetCat `  U
)
31, 2domcatsetval2 25929 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.dom  `  F )  e.  U
)
4 cmpidmor3.5 . . . . . 6  |- .id  =  ( Id SetCat `  U
)
52, 4domidmor2 25949 . . . . 5  |-  ( ( U  e.  Univ  /\  (.dom  `  F )  e.  U
)  ->  (.dom  `  (.id  `  (.dom  `  F ) ) )  =  (.dom  `  F ) )
63, 5syldan 456 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.dom  `  (.id  `  (.dom  `  F ) ) )  =  (.dom  `  F ) )
76opeq1d 3802 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. (.dom  `  (.id  `  (.dom  `  F ) ) ) ,  ( ( cod SetCat `
 U ) `  F ) >.  =  <. (.dom  `  F ) ,  ( ( cod SetCat `
 U ) `  F ) >. )
8 eqid 2283 . . . . . . 7  |-  ( graph SetCat `  U )  =  (
graph SetCat `  U )
98, 4grphidmor3 25954 . . . . . 6  |-  ( ( U  e.  Univ  /\  (.dom  `  F )  e.  U
)  ->  ( ( graph
SetCat `  U ) `  (.id  `  (.dom  `  F ) ) )  =  (  _I  |`  (.dom  `  F ) ) )
103, 9syldan 456 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( graph SetCat `  U ) `  (.id  `  (.dom  `  F ) ) )  =  (  _I  |`  (.dom  `  F ) ) )
1110coeq2d 4846 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( ( graph SetCat `  U
) `  F )  o.  ( ( graph SetCat `  U
) `  (.id  `  (.dom  `  F
) ) ) )  =  ( ( (
graph SetCat `  U ) `  F )  o.  (  _I  |`  (.dom  `  F ) ) ) )
12 eqid 2283 . . . . . . 7  |-  ( cod SetCat `
 U )  =  ( cod SetCat `  U
)
132, 12, 8, 1prismorcset3 25938 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( graph SetCat `  U ) `  F )  e.  ( ( ( cod SetCat `  U
) `  F )  ^m  (.dom  `  F
) ) )
14 fvex 5539 . . . . . . . 8  |-  ( ( cod SetCat `  U ) `  F )  e.  _V
15 fvex 5539 . . . . . . . 8  |-  (.dom  `  F )  e.  _V
1614, 15pm3.2i 441 . . . . . . 7  |-  ( ( ( cod SetCat `  U
) `  F )  e.  _V  /\  (.dom  `  F )  e.  _V )
17 elmapg 6785 . . . . . . 7  |-  ( ( ( ( cod SetCat `  U
) `  F )  e.  _V  /\  (.dom  `  F )  e.  _V )  ->  ( ( (
graph SetCat `  U ) `  F )  e.  ( ( ( cod SetCat `  U
) `  F )  ^m  (.dom  `  F
) )  <->  ( ( graph
SetCat `  U ) `  F ) : (.dom  `  F ) --> ( ( cod SetCat `  U
) `  F )
) )
1816, 17mp1i 11 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( ( graph SetCat `  U
) `  F )  e.  ( ( ( cod SetCat `
 U ) `  F )  ^m  (.dom  `  F ) )  <->  ( ( graph
SetCat `  U ) `  F ) : (.dom  `  F ) --> ( ( cod SetCat `  U
) `  F )
) )
1913, 18mpbid 201 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( graph SetCat `  U ) `  F ) : (.dom  `  F ) --> ( ( cod SetCat `  U
) `  F )
)
20 fcoi1 5415 . . . . 5  |-  ( ( ( graph SetCat `  U ) `  F ) : (.dom  `  F ) --> ( ( cod SetCat `  U
) `  F )  ->  ( ( ( graph SetCat `  U ) `  F
)  o.  (  _I  |`  (.dom  `  F
) ) )  =  ( ( graph SetCat `  U
) `  F )
)
2119, 20syl 15 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( ( graph SetCat `  U
) `  F )  o.  (  _I  |`  (.dom  `  F ) ) )  =  ( ( graph SetCat `  U ) `  F
) )
2211, 21eqtrd 2315 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( ( graph SetCat `  U
) `  F )  o.  ( ( graph SetCat `  U
) `  (.id  `  (.dom  `  F
) ) ) )  =  ( ( graph SetCat `  U ) `  F
) )
237, 22opeq12d 3804 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. <. (.dom  `  (.id  `  (.dom  `  F ) ) ) ,  ( ( cod SetCat `
 U ) `  F ) >. ,  ( ( ( graph SetCat `  U
) `  F )  o.  ( ( graph SetCat `  U
) `  (.id  `  (.dom  `  F
) ) ) )
>.  =  <. <. (.dom  `  F ) ,  ( ( cod SetCat `  U
) `  F ) >. ,  ( ( graph SetCat `  U ) `  F
) >. )
24 simpl 443 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  U  e.  Univ )
25 simpr 447 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  e. .Morphism  )
264, 1idcatval2 25944 . . . 4  |-  ( ( U  e.  Univ  /\  (.dom  `  F )  e.  U
)  ->  (.id  `  (.dom  `  F
) )  e. .Morphism  )
273, 26syldan 456 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.id  `  (.dom  `  F
) )  e. .Morphism  )
2812, 4codidmor2 25951 . . . . 5  |-  ( ( U  e.  Univ  /\  (.dom  `  F )  e.  U
)  ->  ( ( cod
SetCat `  U ) `  (.id  `  (.dom  `  F ) ) )  =  (.dom  `  F ) )
293, 28syldan 456 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( cod SetCat `  U
) `  (.id  `  (.dom  `  F
) ) )  =  (.dom  `  F
) )
3029eqcomd 2288 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.dom  `  F )  =  ( ( cod SetCat `  U
) `  (.id  `  (.dom  `  F
) ) ) )
31 cmpidmor3.1 . . . 4  |-  O  =  ( ro SetCat `  U
)
3231, 1, 2, 12, 8cmp2morpcatt 25962 . . 3  |-  ( ( U  e.  Univ  /\  ( F  e. .Morphism  /\  (.id  `  (.dom  `  F
) )  e. .Morphism  )  /\  (.dom  `  F )  =  ( ( cod SetCat `  U
) `  (.id  `  (.dom  `  F
) ) ) )  ->  ( F O (.id  `  (.dom  `  F ) ) )  =  <. <. (.dom  `  (.id  `  (.dom  `  F ) ) ) ,  ( ( cod SetCat `
 U ) `  F ) >. ,  ( ( ( graph SetCat `  U
) `  F )  o.  ( ( graph SetCat `  U
) `  (.id  `  (.dom  `  F
) ) ) )
>. )
3324, 25, 27, 30, 32syl121anc 1187 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( F O (.id  `  (.dom  `  F
) ) )  = 
<. <. (.dom  `  (.id  `  (.dom  `  F ) ) ) ,  ( ( cod SetCat `
 U ) `  F ) >. ,  ( ( ( graph SetCat `  U
) `  F )  o.  ( ( graph SetCat `  U
) `  (.id  `  (.dom  `  F
) ) ) )
>. )
341, 2, 12, 8morexcmp2 25968 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  ( ( cod SetCat `  U
) `  F ) >. ,  ( ( graph SetCat `  U ) `  F
) >. )
3523, 33, 343eqtr4d 2325 1  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( F O (.id  `  (.dom  `  F
) ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   graph SetCatcgraphcase 25921   cod
SetCatccodcase 25932   Id SetCatcidcase 25939   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-idcatset 25940  df-rocatset 25956
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