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Theorem cmpidmor2 26072
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 5542 . . . . . 6  |-  (.id  `  (.cod  `  F
) )  =  ( ( Id SetCat `  U
) `  (.cod  `  F ) )
41, 3fveq12i 5546 . . . . 5  |-  (.cod  `  (.id  `  (.cod  `  F ) ) )  =  ( ( cod SetCat `
 U ) `  ( ( Id SetCat `  U ) `  (.cod  `  F ) ) )
51fveq1i 5542 . . . . . . 7  |-  (.cod  `  F )  =  ( ( cod SetCat `  U
) `  F )
6 cmpidmor2.2 . . . . . . . . 9  |- .Morphism  =  ( Morphism SetCat `  U )
76eleq2i 2360 . . . . . . . 8  |-  ( F  e. .Morphism  <->  F  e.  ( Morphism SetCat `  U )
)
8 codcatsetval 26038 . . . . . . . 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 2380 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.cod  `  F )  e.  U
)
11 codidmor 26053 . . . . . 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 2340 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.cod  `  (.id  `  (.cod  `  F ) ) )  =  (.cod  `  F ) )
1413opeq2d 3819 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  <. (
( dom SetCat `  U
) `  F ) ,  (.cod  `  (.id  `  (.cod  `  F
) ) ) >.  =  <. ( ( dom SetCat `
 U ) `  F ) ,  (.cod  `  F )
>. )
153fveq2i 5544 . . . . . 6  |-  ( 2nd `  (.id  `  (.cod  `  F ) ) )  =  ( 2nd `  (
( Id SetCat `  U
) `  (.cod  `  F ) ) )
16 eqid 2296 . . . . . . . 8  |-  2nd  =  2nd
17 eqid 2296 . . . . . . . 8  |-  ( Id SetCat `
 U )  =  ( Id SetCat `  U
)
1816, 17grphidmor2 26056 . . . . . . 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 2340 . . . . 5  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  (.id  `  (.cod  `  F
) ) )  =  (  _I  |`  (.cod  `  F ) ) )
2120coeq1d 4861 . . . 4  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( 2nd `  (.id  `  (.cod  `  F
) ) )  o.  ( 2nd `  F
) )  =  ( (  _I  |`  (.cod  `  F ) )  o.  ( 2nd `  F
) ) )
22 isgraphmrph 26026 . . . . . . . 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 2296 . . . . . . . 8  |-  ( dom SetCat `
 U )  =  ( dom SetCat `  U
)
25 eqid 2296 . . . . . . . 8  |-  ( graph SetCat `  U )  =  (
graph SetCat `  U )
2624, 1, 25, 6prismorcset3 26041 . . . . . . 7  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( graph SetCat `  U ) `  F )  e.  ( (.cod  `  F
)  ^m  ( ( dom
SetCat `  U ) `  F ) ) )
2723, 26eqeltrrd 2371 . . . . . 6  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( 2nd `  F )  e.  ( (.cod  `  F )  ^m  (
( dom SetCat `  U
) `  F )
) )
28 fvex 5555 . . . . . . 7  |-  (.cod  `  F )  e.  _V
29 fvex 5555 . . . . . . 7  |-  ( ( dom SetCat `  U ) `  F )  e.  _V
3028, 29elmap 6812 . . . . . 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 5432 . . . . 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 2328 . . 3  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (
( 2nd `  (.id  `  (.cod  `  F
) ) )  o.  ( 2nd `  F
) )  =  ( 2nd `  F ) )
3514, 34opeq12d 3820 . 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 26047 . . . 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 5544 . . . . 5  |-  ( ( dom SetCat `  U ) `  (.id  `  (.cod  `  F ) ) )  =  ( ( dom SetCat `
 U ) `  ( ( Id SetCat `  U ) `  (.cod  `  F ) ) )
41 domidmor 26051 . . . . 5  |-  ( ( U  e.  Univ  /\  (.cod  `  F )  e.  U
)  ->  ( ( dom
SetCat `  U ) `  ( ( Id SetCat `  U ) `  (.cod  `  F ) ) )  =  (.cod  `  F ) )
4240, 41syl5eq 2340 . . . 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 26064 . . 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 26070 . 2  |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. ( ( dom SetCat `
 U ) `  F ) ,  (.cod  `  F )
>. ,  ( 2nd `  F ) >. )
4835, 46, 473eqtr4d 2338 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 1632    e. wcel 1696   <.cop 3656    _I cid 4320    |` cres 4707    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137    ^m cmap 6788   Univcgru 8428   Morphism SetCatccmrcase 26013   dom
SetCatcdomcase 26022   graph SetCatcgraphcase 26024   cod
SetCatccodcase 26035   Id SetCatcidcase 26042   ro SetCatcrocase 26058
This theorem is referenced by:  setiscat  26082
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-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-morcatset 26014  df-domcatset 26023  df-graphcatset 26025  df-codcatset 26036  df-idcatset 26043  df-rocatset 26059
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