Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  morexcmp2 Unicode version

Theorem morexcmp2 26071
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 2296 . . 3  |-  2nd  =  2nd
51, 2, 3, 4morexcmp 26070 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  (.cod  `  F )
>. ,  ( 2nd `  F ) >. )
6 morexcmp2.4 . . . . 5  |- .graph  =  ( graph SetCat `  U )
76, 1isgraphmrph2 26027 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.graph  `  F )  =  ( 2nd `  F ) )
87eqcomd 2301 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  F )  =  (.graph  `  F
) )
98opeq2d 3819 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. <. (.dom  `  F ) ,  (.cod  `  F )
>. ,  ( 2nd `  F ) >.  =  <. <.
(.dom  `  F
) ,  (.cod  `  F ) >. ,  (.graph  `  F )
>. )
105, 9eqtrd 2328 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 1632    e. wcel 1696   <.cop 3656   ` cfv 5271   2ndc2nd 6137   Univcgru 8428   Morphism SetCatccmrcase 26013   dom
SetCatcdomcase 26022   graph SetCatcgraphcase 26024   cod
SetCatccodcase 26035
This theorem is referenced by:  cmpidmor3  26073
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-1st 6138  df-2nd 6139  df-morcatset 26014  df-domcatset 26023  df-graphcatset 26025  df-codcatset 26036
  Copyright terms: Public domain W3C validator