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Theorem morexcmp2 25968
Description: A morphism expressed thanks to its components. (Contributed by FL, 8-Nov-2013.)
Hypotheses
Ref Expression
morexcmp2.1  |- .Morphism  =  ( Morphism SetCat `  U )
morexcmp2.2  |- .dom  =  ( dom SetCat `  U
)
morexcmp2.3  |- .cod  =  ( cod SetCat `  U
)
morexcmp2.4  |- .graph  =  ( graph SetCat `  U )
Assertion
Ref Expression
morexcmp2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  (.cod  `  F )
>. ,  (.graph  `  F ) >. )

Proof of Theorem morexcmp2
StepHypRef Expression
1 morexcmp2.1 . . 3  |- .Morphism  =  ( Morphism SetCat `  U )
2 morexcmp2.2 . . 3  |- .dom  =  ( dom SetCat `  U
)
3 morexcmp2.3 . . 3  |- .cod  =  ( cod SetCat `  U
)
4 eqid 2283 . . 3  |-  2nd  =  2nd
51, 2, 3, 4morexcmp 25967 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  (.cod  `  F )
>. ,  ( 2nd `  F ) >. )
6 morexcmp2.4 . . . . 5  |- .graph  =  ( graph SetCat `  U )
76, 1isgraphmrph2 25924 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.graph  `  F )  =  ( 2nd `  F ) )
87eqcomd 2288 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  F )  =  (.graph  `  F
) )
98opeq2d 3803 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. <. (.dom  `  F ) ,  (.cod  `  F )
>. ,  ( 2nd `  F ) >.  =  <. <.
(.dom  `  F
) ,  (.cod  `  F ) >. ,  (.graph  `  F )
>. )
105, 9eqtrd 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    = wceq 1623    e. wcel 1684   <.cop 3643   ` cfv 5255   2ndc2nd 6121   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   graph SetCatcgraphcase 25921   cod
SetCatccodcase 25932
This theorem is referenced by:  cmpidmor3  25970
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-graphcatset 25922  df-codcatset 25933
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