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Theorem cmpidmor3 26073
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 26032 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.dom  `  F )  e.  U
)
4 cmpidmor3.5 . . . . . 6  |- .id  =  ( Id SetCat `  U
)
52, 4domidmor2 26052 . . . . 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 3818 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. (.dom  `  (.id  `  (.dom  `  F ) ) ) ,  ( ( cod SetCat `
 U ) `  F ) >.  =  <. (.dom  `  F ) ,  ( ( cod SetCat `
 U ) `  F ) >. )
8 eqid 2296 . . . . . . 7  |-  ( graph SetCat `  U )  =  (
graph SetCat `  U )
98, 4grphidmor3 26057 . . . . . 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 4862 . . . 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 2296 . . . . . . 7  |-  ( cod SetCat `
 U )  =  ( cod SetCat `  U
)
132, 12, 8, 1prismorcset3 26041 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( graph SetCat `  U ) `  F )  e.  ( ( ( cod SetCat `  U
) `  F )  ^m  (.dom  `  F
) ) )
14 fvex 5555 . . . . . . . 8  |-  ( ( cod SetCat `  U ) `  F )  e.  _V
15 fvex 5555 . . . . . . . 8  |-  (.dom  `  F )  e.  _V
1614, 15pm3.2i 441 . . . . . . 7  |-  ( ( ( cod SetCat `  U
) `  F )  e.  _V  /\  (.dom  `  F )  e.  _V )
17 elmapg 6801 . . . . . . 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 5431 . . . . 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 2328 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( ( graph SetCat `  U
) `  F )  o.  ( ( graph SetCat `  U
) `  (.id  `  (.dom  `  F
) ) ) )  =  ( ( graph SetCat `  U ) `  F
) )
237, 22opeq12d 3820 . 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 26047 . . . 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 26054 . . . . 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 2301 . . 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 26065 . . 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 26071 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  ( ( cod SetCat `  U
) `  F ) >. ,  ( ( graph SetCat `  U ) `  F
) >. )
3523, 33, 343eqtr4d 2338 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    _I cid 4320    |` cres 4707    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Univcgru 8428   Morphism SetCatccmrcase 26013   dom
SetCatcdomcase 26022   graph SetCatcgraphcase 26024   cod
SetCatccodcase 26035   Id SetCatcidcase 26042   ro SetCatcrocase 26058
This theorem is referenced by:  setiscat  26082
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-idcatset 26043  df-rocatset 26059
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