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Theorem cmpidmor2 25969
Description: Composition with an identity. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
cmpidmor2.1  |-  O  =  ( ro SetCat `  U
)
cmpidmor2.2  |- .Morphism  =  ( Morphism SetCat `  U )
cmpidmor2.4  |- .cod  =  ( cod SetCat `  U
)
cmpidmor2.5  |- .id  =  ( Id SetCat `  U
)
Assertion
Ref Expression
cmpidmor2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
(.id  `  (.cod  `  F ) ) O F )  =  F )

Proof of Theorem cmpidmor2
StepHypRef Expression
1 cmpidmor2.4 . . . . . 6  |- .cod  =  ( cod SetCat `  U
)
2 cmpidmor2.5 . . . . . . 7  |- .id  =  ( Id SetCat `  U
)
32fveq1i 5526 . . . . . 6  |-  (.id  `  (.cod  `  F
) )  =  ( ( Id SetCat `  U
) `  (.cod  `  F ) )
41, 3fveq12i 5530 . . . . 5  |-  (.cod  `  (.id  `  (.cod  `  F ) ) )  =  ( ( cod SetCat `
 U ) `  ( ( Id SetCat `  U ) `  (.cod  `  F ) ) )
51fveq1i 5526 . . . . . . 7  |-  (.cod  `  F )  =  ( ( cod SetCat `  U
) `  F )
6 cmpidmor2.2 . . . . . . . . 9  |- .Morphism  =  ( Morphism SetCat `  U )
76eleq2i 2347 . . . . . . . 8  |-  ( F  e. .Morphism  <->  F  e.  ( Morphism SetCat `  U )
)
8 codcatsetval 25935 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  F  e.  ( Morphism SetCat `  U )
)  ->  ( ( cod
SetCat `  U ) `  F )  e.  U
)
97, 8sylan2b 461 . . . . . . 7  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( cod SetCat `  U
) `  F )  e.  U )
105, 9syl5eqel 2367 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.cod  `  F )  e.  U
)
11 codidmor 25950 . . . . . 6  |-  ( ( U  e.  Univ  /\  (.cod  `  F )  e.  U
)  ->  ( ( cod
SetCat `  U ) `  ( ( Id SetCat `  U ) `  (.cod  `  F ) ) )  =  (.cod  `  F ) )
1210, 11syldan 456 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( cod SetCat `  U
) `  ( ( Id SetCat `  U ) `  (.cod  `  F
) ) )  =  (.cod  `  F
) )
134, 12syl5eq 2327 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.cod  `  (.id  `  (.cod  `  F ) ) )  =  (.cod  `  F ) )
1413opeq2d 3803 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. (
( dom SetCat `  U
) `  F ) ,  (.cod  `  (.id  `  (.cod  `  F
) ) ) >.  =  <. ( ( dom SetCat `
 U ) `  F ) ,  (.cod  `  F )
>. )
153fveq2i 5528 . . . . . 6  |-  ( 2nd `  (.id  `  (.cod  `  F ) ) )  =  ( 2nd `  (
( Id SetCat `  U
) `  (.cod  `  F ) ) )
16 eqid 2283 . . . . . . . 8  |-  2nd  =  2nd
17 eqid 2283 . . . . . . . 8  |-  ( Id SetCat `
 U )  =  ( Id SetCat `  U
)
1816, 17grphidmor2 25953 . . . . . . 7  |-  ( ( U  e.  Univ  /\  (.cod  `  F )  e.  U
)  ->  ( 2nd `  ( ( Id SetCat `  U ) `  (.cod  `  F ) ) )  =  (  _I  |`  (.cod  `  F ) ) )
1910, 18syldan 456 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  ( ( Id SetCat `
 U ) `  (.cod  `  F
) ) )  =  (  _I  |`  (.cod  `  F ) ) )
2015, 19syl5eq 2327 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  (.id  `  (.cod  `  F
) ) )  =  (  _I  |`  (.cod  `  F ) ) )
2120coeq1d 4845 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( 2nd `  (.id  `  (.cod  `  F
) ) )  o.  ( 2nd `  F
) )  =  ( (  _I  |`  (.cod  `  F ) )  o.  ( 2nd `  F
) ) )
22 isgraphmrph 25923 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  F  e.  ( Morphism SetCat `  U )
)  ->  ( ( graph
SetCat `  U ) `  F )  =  ( 2nd `  F ) )
237, 22sylan2b 461 . . . . . . 7  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( graph SetCat `  U ) `  F )  =  ( 2nd `  F ) )
24 eqid 2283 . . . . . . . 8  |-  ( dom SetCat `
 U )  =  ( dom SetCat `  U
)
25 eqid 2283 . . . . . . . 8  |-  ( graph SetCat `  U )  =  (
graph SetCat `  U )
2624, 1, 25, 6prismorcset3 25938 . . . . . . 7  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( graph SetCat `  U ) `  F )  e.  ( (.cod  `  F
)  ^m  ( ( dom
SetCat `  U ) `  F ) ) )
2723, 26eqeltrrd 2358 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  F )  e.  ( (.cod  `  F )  ^m  (
( dom SetCat `  U
) `  F )
) )
28 fvex 5539 . . . . . . 7  |-  (.cod  `  F )  e.  _V
29 fvex 5539 . . . . . . 7  |-  ( ( dom SetCat `  U ) `  F )  e.  _V
3028, 29elmap 6796 . . . . . 6  |-  ( ( 2nd `  F )  e.  ( (.cod  `  F )  ^m  (
( dom SetCat `  U
) `  F )
)  <->  ( 2nd `  F
) : ( ( dom SetCat `  U ) `  F ) --> (.cod  `  F ) )
3127, 30sylib 188 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  F ) : ( ( dom SetCat `  U
) `  F ) --> (.cod  `  F
) )
32 fcoi2 5416 . . . . 5  |-  ( ( 2nd `  F ) : ( ( dom SetCat `
 U ) `  F ) --> (.cod  `  F )  ->  (
(  _I  |`  (.cod  `  F ) )  o.  ( 2nd `  F
) )  =  ( 2nd `  F ) )
3331, 32syl 15 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
(  _I  |`  (.cod  `  F ) )  o.  ( 2nd `  F
) )  =  ( 2nd `  F ) )
3421, 33eqtrd 2315 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( 2nd `  (.id  `  (.cod  `  F
) ) )  o.  ( 2nd `  F
) )  =  ( 2nd `  F ) )
3514, 34opeq12d 3804 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. <. (
( dom SetCat `  U
) `  F ) ,  (.cod  `  (.id  `  (.cod  `  F
) ) ) >. ,  ( ( 2nd `  (.id  `  (.cod  `  F ) ) )  o.  ( 2nd `  F
) ) >.  =  <. <.
( ( dom SetCat `  U
) `  F ) ,  (.cod  `  F
) >. ,  ( 2nd `  F ) >. )
36 simpl 443 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  U  e.  Univ )
372, 6idcatval2 25944 . . . 4  |-  ( ( U  e.  Univ  /\  (.cod  `  F )  e.  U
)  ->  (.id  `  (.cod  `  F
) )  e. .Morphism  )
3810, 37syldan 456 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.id  `  (.cod  `  F
) )  e. .Morphism  )
39 simpr 447 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  e. .Morphism  )
403fveq2i 5528 . . . . 5  |-  ( ( dom SetCat `  U ) `  (.id  `  (.cod  `  F ) ) )  =  ( ( dom SetCat `
 U ) `  ( ( Id SetCat `  U ) `  (.cod  `  F ) ) )
41 domidmor 25948 . . . . 5  |-  ( ( U  e.  Univ  /\  (.cod  `  F )  e.  U
)  ->  ( ( dom
SetCat `  U ) `  ( ( Id SetCat `  U ) `  (.cod  `  F ) ) )  =  (.cod  `  F ) )
4240, 41syl5eq 2327 . . . 4  |-  ( ( U  e.  Univ  /\  (.cod  `  F )  e.  U
)  ->  ( ( dom
SetCat `  U ) `  (.id  `  (.cod  `  F ) ) )  =  (.cod  `  F ) )
4310, 42syldan 456 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( dom SetCat `  U
) `  (.id  `  (.cod  `  F
) ) )  =  (.cod  `  F
) )
44 cmpidmor2.1 . . . 4  |-  O  =  ( ro SetCat `  U
)
4544, 6, 24, 1cmp2morpcats 25961 . . 3  |-  ( ( U  e.  Univ  /\  (
(.id  `  (.cod  `  F ) )  e. .Morphism  /\  F  e. .Morphism  )  /\  ( ( dom SetCat `  U
) `  (.id  `  (.cod  `  F
) ) )  =  (.cod  `  F
) )  ->  (
(.id  `  (.cod  `  F ) ) O F )  =  <. <.
( ( dom SetCat `  U
) `  F ) ,  (.cod  `  (.id  `  (.cod  `  F
) ) ) >. ,  ( ( 2nd `  (.id  `  (.cod  `  F ) ) )  o.  ( 2nd `  F
) ) >. )
4636, 38, 39, 43, 45syl121anc 1187 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
(.id  `  (.cod  `  F ) ) O F )  =  <. <.
( ( dom SetCat `  U
) `  F ) ,  (.cod  `  (.id  `  (.cod  `  F
) ) ) >. ,  ( ( 2nd `  (.id  `  (.cod  `  F ) ) )  o.  ( 2nd `  F
) ) >. )
476, 24, 1, 16morexcmp 25967 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. ( ( dom SetCat `
 U ) `  F ) ,  (.cod  `  F )
>. ,  ( 2nd `  F ) >. )
4835, 46, 473eqtr4d 2325 1  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
(.id  `  (.cod  `  F ) ) O F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   2ndc2nd 6121    ^m cmap 6772   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   graph SetCatcgraphcase 25921   cod
SetCatccodcase 25932   Id SetCatcidcase 25939   ro SetCatcrocase 25955
This theorem is referenced by:  setiscat  25979
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-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-morcatset 25911  df-domcatset 25920  df-graphcatset 25922  df-codcatset 25933  df-idcatset 25940  df-rocatset 25956
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