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Theorem grphidmor 25952
Description: Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
grphidmor  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( graph SetCat `  U ) `  ( ( Id SetCat `  U ) `  A
) )  =  (  _I  |`  A )
)

Proof of Theorem grphidmor
StepHypRef Expression
1 idmor 25946 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( Id SetCat `  U
) `  A )  =  <. <. A ,  A >. ,  (  _I  |`  A )
>. )
21fveq2d 5529 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( graph SetCat `  U ) `  ( ( Id SetCat `  U ) `  A
) )  =  ( ( graph SetCat `  U ) `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )
3 idmorimor 25947 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  <. <. A ,  A >. ,  (  _I  |`  A ) >.  e.  (
Morphism
SetCat `  U ) )
4 isgraphmrph 25923 . . 3  |-  ( ( U  e.  Univ  /\  <. <. A ,  A >. ,  (  _I  |`  A )
>.  e.  ( Morphism SetCat `  U
) )  ->  (
( graph SetCat `  U ) `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )
53, 4syldan 456 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( graph SetCat `  U ) `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )
6 opex 4237 . . 3  |-  <. A ,  A >.  e.  _V
7 funi 5284 . . . 4  |-  Fun  _I
8 elex 2796 . . . . 5  |-  ( A  e.  U  ->  A  e.  _V )
98adantl 452 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  e.  _V )
10 resfunexg 5737 . . . 4  |-  ( ( Fun  _I  /\  A  e.  _V )  ->  (  _I  |`  A )  e. 
_V )
117, 9, 10sylancr 644 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (  _I  |`  A )  e. 
_V )
12 op2ndg 6133 . . 3  |-  ( (
<. A ,  A >.  e. 
_V  /\  (  _I  |`  A )  e.  _V )  ->  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  (  _I  |`  A ) )
136, 11, 12sylancr 644 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  (  _I  |`  A ) )
142, 5, 133eqtrd 2319 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( graph SetCat `  U ) `  ( ( Id SetCat `  U ) `  A
) )  =  (  _I  |`  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    _I cid 4304    |` cres 4691   Fun wfun 5249   ` cfv 5255   2ndc2nd 6121   Univcgru 8412   Morphism SetCatccmrcase 25910   graph SetCatcgraphcase 25921   Id SetCatcidcase 25939
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-2nd 6123  df-map 6774  df-morcatset 25911  df-graphcatset 25922  df-idcatset 25940
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