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Theorem prismorcset3 25938
Description: The predicate "is a morphism of the category Set". (Contributed by FL, 6-Nov-2013.)
Hypotheses
Ref Expression
prismorcset3.1  |- .dom  =  ( dom SetCat `  U
)
prismorcset3.2  |- .cod  =  ( cod SetCat `  U
)
prismorcset3.3  |- .graph  =  ( graph SetCat `  U )
prismorcset3.4  |- .Morphism  =  ( Morphism SetCat `  U )
Assertion
Ref Expression
prismorcset3  |-  ( ( U  e.  Univ  /\  M  e. .Morphism  )  ->  (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  (.dom  `  M ) ) )

Proof of Theorem prismorcset3
StepHypRef Expression
1 prismorcset3.4 . 2  |- .Morphism  =  ( Morphism SetCat `  U )
2 eleq2 2344 . . . 4  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( M  e. .Morphism  <->  M  e.  ( Morphism SetCat `  U )
) )
32anbi2d 684 . . 3  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( U  e. 
Univ  /\  M  e. .Morphism  ) 
<->  ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U ) ) ) )
4 prismorcset3.1 . . . 4  |- .dom  =  ( dom SetCat `  U
)
5 prismorcset3.2 . . . . . 6  |- .cod  =  ( cod SetCat `  U
)
6 prismorcset3.3 . . . . . . . 8  |- .graph  =  ( graph SetCat `  U )
7 eqid 2283 . . . . . . . . . . . 12  |-  ( ( 1st  o.  1st ) `  M )  =  ( ( 1st  o.  1st ) `  M )
8 eqid 2283 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  1st ) `  M )  =  ( ( 2nd  o.  1st ) `  M )
9 eqid 2283 . . . . . . . . . . . 12  |-  ( 2nd `  M )  =  ( 2nd `  M )
107, 8, 9prismorcset2 25918 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( (
( 1st  o.  1st ) `  M )  e.  U  /\  (
( 2nd  o.  1st ) `  M )  e.  U  /\  ( 2nd `  M )  e.  ( ( ( 2nd 
o.  1st ) `  M
)  ^m  ( ( 1st  o.  1st ) `  M ) ) ) )
1110simp3d 969 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( 2nd `  M )  e.  ( ( ( 2nd  o.  1st ) `  M )  ^m  ( ( 1st 
o.  1st ) `  M
) ) )
12 isgraphmrph 25923 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( ( graph
SetCat `  U ) `  M )  =  ( 2nd `  M ) )
13 codcatval 25936 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( ( cod
SetCat `  U ) `  M )  =  ( ( 2nd  o.  1st ) `  M )
)
14 domcatval 25930 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  M )  =  ( ( 1st  o.  1st ) `  M )
)
1513, 14oveq12d 5876 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( (
( cod SetCat `  U
) `  M )  ^m  ( ( dom SetCat `  U
) `  M )
)  =  ( ( ( 2nd  o.  1st ) `  M )  ^m  ( ( 1st  o.  1st ) `  M ) ) )
1612, 15eleq12d 2351 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( (
( graph SetCat `  U ) `  M )  e.  ( ( ( cod SetCat `  U
) `  M )  ^m  ( ( dom SetCat `  U
) `  M )
)  <->  ( 2nd `  M
)  e.  ( ( ( 2nd  o.  1st ) `  M )  ^m  ( ( 1st  o.  1st ) `  M ) ) ) )
1711, 16mpbird 223 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  ( ( graph
SetCat `  U ) `  M )  e.  ( ( ( cod SetCat `  U
) `  M )  ^m  ( ( dom SetCat `  U
) `  M )
) )
18 fveq1 5524 . . . . . . . . . 10  |-  (.graph  =  ( graph SetCat `  U
)  ->  (.graph  `  M )  =  ( ( graph SetCat `  U ) `  M ) )
1918eleq1d 2349 . . . . . . . . 9  |-  (.graph  =  ( graph SetCat `  U
)  ->  ( (.graph  `  M )  e.  ( ( ( cod SetCat `  U
) `  M )  ^m  ( ( dom SetCat `  U
) `  M )
)  <->  ( ( graph SetCat `  U ) `  M
)  e.  ( ( ( cod SetCat `  U
) `  M )  ^m  ( ( dom SetCat `  U
) `  M )
) ) )
2017, 19syl5ibr 212 . . . . . . . 8  |-  (.graph  =  ( graph SetCat `  U
)  ->  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  (.graph  `  M )  e.  ( ( ( cod SetCat `  U
) `  M )  ^m  ( ( dom SetCat `  U
) `  M )
) ) )
216, 20ax-mp 8 . . . . . . 7  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  (.graph  `  M )  e.  ( ( ( cod SetCat `  U
) `  M )  ^m  ( ( dom SetCat `  U
) `  M )
) )
22 fveq1 5524 . . . . . . . . 9  |-  (.cod  =  ( cod SetCat `  U
)  ->  (.cod  `  M )  =  ( ( cod SetCat `  U
) `  M )
)
2322oveq1d 5873 . . . . . . . 8  |-  (.cod  =  ( cod SetCat `  U
)  ->  ( (.cod  `  M )  ^m  (
( dom SetCat `  U
) `  M )
)  =  ( ( ( cod SetCat `  U
) `  M )  ^m  ( ( dom SetCat `  U
) `  M )
) )
2423eleq2d 2350 . . . . . . 7  |-  (.cod  =  ( cod SetCat `  U
)  ->  ( (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  ( ( dom
SetCat `  U ) `  M ) )  <->  (.graph  `  M )  e.  ( ( ( cod SetCat `  U
) `  M )  ^m  ( ( dom SetCat `  U
) `  M )
) ) )
2521, 24syl5ibr 212 . . . . . 6  |-  (.cod  =  ( cod SetCat `  U
)  ->  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  ( ( dom
SetCat `  U ) `  M ) ) ) )
265, 25ax-mp 8 . . . . 5  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  ( ( dom
SetCat `  U ) `  M ) ) )
27 fveq1 5524 . . . . . . 7  |-  (.dom  =  ( dom SetCat `  U
)  ->  (.dom  `  M )  =  ( ( dom SetCat `  U
) `  M )
)
2827oveq2d 5874 . . . . . 6  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( (.cod  `  M )  ^m  (.dom  `  M ) )  =  ( (.cod  `  M )  ^m  (
( dom SetCat `  U
) `  M )
) )
2928eleq2d 2350 . . . . 5  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  (.dom  `  M ) )  <->  (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  ( ( dom
SetCat `  U ) `  M ) ) ) )
3026, 29syl5ibr 212 . . . 4  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  (.dom  `  M ) ) ) )
314, 30ax-mp 8 . . 3  |-  ( ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U )
)  ->  (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  (.dom  `  M ) ) )
323, 31syl6bi 219 . 2  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( U  e. 
Univ  /\  M  e. .Morphism  )  ->  (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  (.dom  `  M ) ) ) )
331, 32ax-mp 8 1  |-  ( ( U  e.  Univ  /\  M  e. .Morphism  )  ->  (.graph  `  M )  e.  ( (.cod  `  M
)  ^m  (.dom  `  M ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    o. ccom 4693   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   graph SetCatcgraphcase 25921   cod
SetCatccodcase 25932
This theorem is referenced by:  cmpidmor2  25969  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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