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Theorem morexcmp 25967
Description: A morphism expressed thanks to its components. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
morexcmp.1  |- .Morphism  =  ( Morphism SetCat `  U )
morexcmp.2  |- .dom  =  ( dom SetCat `  U
)
morexcmp.3  |- .cod  =  ( cod SetCat `  U
)
morexcmp.4  |- .graph  =  2nd
Assertion
Ref Expression
morexcmp  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  (.cod  `  F )
>. ,  (.graph  `  F ) >. )

Proof of Theorem morexcmp
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 morexcmp.1 . . . . . 6  |- .Morphism  =  ( Morphism SetCat `  U )
2 morcatset1 25915 . . . . . 6  |-  ( U  e.  Univ  ->  ( Morphism SetCat `  U )  =  { <. <. a ,  b
>. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  (
b  ^m  a )
) } )
31, 2syl5eq 2327 . . . . 5  |-  ( U  e.  Univ  -> .Morphism  =  { <. <. a ,  b
>. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  (
b  ^m  a )
) } )
4 oprabss 5933 . . . . . 6  |-  { <. <.
a ,  b >. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a ) ) } 
C_  ( ( _V 
X.  _V )  X.  _V )
54a1i 10 . . . . 5  |-  ( U  e.  Univ  ->  { <. <.
a ,  b >. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a ) ) } 
C_  ( ( _V 
X.  _V )  X.  _V ) )
63, 5eqsstrd 3212 . . . 4  |-  ( U  e.  Univ  -> .Morphism  C_  ( ( _V  X.  _V )  X.  _V )
)
76sselda 3180 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  e.  ( ( _V  X.  _V )  X.  _V )
)
8 1st2nd2 6159 . . 3  |-  ( F  e.  ( ( _V 
X.  _V )  X.  _V )  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
97, 8syl 15 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
10 xp1st 6149 . . . . 5  |-  ( F  e.  ( ( _V 
X.  _V )  X.  _V )  ->  ( 1st `  F
)  e.  ( _V 
X.  _V ) )
11 1st2nd2 6159 . . . . 5  |-  ( ( 1st `  F )  e.  ( _V  X.  _V )  ->  ( 1st `  F )  =  <. ( 1st `  ( 1st `  F ) ) ,  ( 2nd `  ( 1st `  F ) )
>. )
127, 10, 113syl 18 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 1st `  F )  = 
<. ( 1st `  ( 1st `  F ) ) ,  ( 2nd `  ( 1st `  F ) )
>. )
13 morexcmp.2 . . . . . . 7  |- .dom  =  ( dom SetCat `  U
)
141, 13domcatval2 25931 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.dom  `  F )  =  ( ( 1st  o.  1st ) `  F )
)
15 fo1st 6139 . . . . . . . 8  |-  1st : _V -onto-> _V
16 fof 5451 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
1715, 16ax-mp 8 . . . . . . 7  |-  1st : _V
--> _V
18 elex 2796 . . . . . . . 8  |-  ( F  e. .Morphism  ->  F  e.  _V )
1918adantl 452 . . . . . . 7  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  e.  _V )
20 fvco3 5596 . . . . . . 7  |-  ( ( 1st : _V --> _V  /\  F  e.  _V )  ->  ( ( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
2117, 19, 20sylancr 644 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
2214, 21eqtr2d 2316 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 1st `  ( 1st `  F
) )  =  (.dom  `  F ) )
23 morexcmp.3 . . . . . . 7  |- .cod  =  ( cod SetCat `  U
)
241, 23codcatval2 25937 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.cod  `  F )  =  ( ( 2nd  o.  1st ) `  F )
)
25 fvco3 5596 . . . . . . 7  |-  ( ( 1st : _V --> _V  /\  F  e.  _V )  ->  ( ( 2nd  o.  1st ) `  F )  =  ( 2nd `  ( 1st `  F ) ) )
2617, 19, 25sylancr 644 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( 2nd  o.  1st ) `  F )  =  ( 2nd `  ( 1st `  F ) ) )
2724, 26eqtr2d 2316 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  ( 1st `  F
) )  =  (.cod  `  F ) )
2822, 27opeq12d 3804 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. ( 1st `  ( 1st `  F
) ) ,  ( 2nd `  ( 1st `  F ) ) >.  =  <. (.dom  `  F ) ,  (.cod  `  F )
>. )
2912, 28eqtrd 2315 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 1st `  F )  = 
<. (.dom  `  F
) ,  (.cod  `  F ) >. )
30 morexcmp.4 . . . . . 6  |- .graph  =  2nd
3130eqcomi 2287 . . . . 5  |-  2nd  = .graph
3231fveq1i 5526 . . . 4  |-  ( 2nd `  F )  =  (.graph  `  F )
3332a1i 10 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  F )  =  (.graph  `  F
) )
3429, 33opeq12d 3804 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. ( 1st `  F ) ,  ( 2nd `  F
) >.  =  <. <. (.dom  `  F ) ,  (.cod  `  F )
>. ,  (.graph  `  F ) >. )
359, 34eqtrd 2315 1  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  (.cod  `  F )
>. ,  (.graph  `  F ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   <.cop 3643    X. cxp 4687    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   {coprab 5859   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   cod
SetCatccodcase 25932
This theorem is referenced by:  morexcmp2  25968  cmpidmor2  25969
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-1st 6122  df-2nd 6123  df-morcatset 25911  df-domcatset 25920  df-codcatset 25933
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