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Theorem domidmor 25948
Description: Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
domidmor  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( dom SetCat `  U
) `  ( ( Id SetCat `  U ) `  A ) )  =  A )

Proof of Theorem domidmor
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 )  ->  (
( dom SetCat `  U
) `  ( ( Id SetCat `  U ) `  A ) )  =  ( ( dom SetCat `  U
) `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )
)
3 idmorimor 25947 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  <. <. A ,  A >. ,  (  _I  |`  A ) >.  e.  (
Morphism
SetCat `  U ) )
4 domcatval 25930 . . . 4  |-  ( ( U  e.  Univ  /\  <. <. A ,  A >. ,  (  _I  |`  A )
>.  e.  ( Morphism SetCat `  U
) )  ->  (
( dom SetCat `  U
) `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )  =  ( ( 1st 
o.  1st ) `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )
53, 4syldan 456 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( dom SetCat `  U
) `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )  =  ( ( 1st 
o.  1st ) `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )
6 fo1st 6139 . . . . . 6  |-  1st : _V -onto-> _V
7 fof 5451 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
86, 7ax-mp 8 . . . . 5  |-  1st : _V
--> _V
9 opex 4237 . . . . 5  |-  <. <. A ,  A >. ,  (  _I  |`  A ) >.  e.  _V
10 fvco3 5596 . . . . 5  |-  ( ( 1st : _V --> _V  /\  <. <. A ,  A >. ,  (  _I  |`  A )
>.  e.  _V )  -> 
( ( 1st  o.  1st ) `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )  =  ( 1st `  ( 1st `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) ) )
118, 9, 10mp2an 653 . . . 4  |-  ( ( 1st  o.  1st ) `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  ( 1st `  ( 1st `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )
12 opex 4237 . . . . . . 7  |-  <. A ,  A >.  e.  _V
13 funi 5284 . . . . . . . 8  |-  Fun  _I
14 simpr 447 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  e.  U )
15 resfunexg 5737 . . . . . . . 8  |-  ( ( Fun  _I  /\  A  e.  U )  ->  (  _I  |`  A )  e. 
_V )
1613, 14, 15sylancr 644 . . . . . . 7  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (  _I  |`  A )  e. 
_V )
17 op1stg 6132 . . . . . . 7  |-  ( (
<. A ,  A >.  e. 
_V  /\  (  _I  |`  A )  e.  _V )  ->  ( 1st `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  <. A ,  A >. )
1812, 16, 17sylancr 644 . . . . . 6  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 1st `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  <. A ,  A >. )
1918fveq2d 5529 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 1st `  ( 1st `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )  =  ( 1st `  <. A ,  A >. ) )
20 op1stg 6132 . . . . . 6  |-  ( ( A  e.  U  /\  A  e.  U )  ->  ( 1st `  <. A ,  A >. )  =  A )
2114, 20sylancom 648 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 1st `  <. A ,  A >. )  =  A )
2219, 21eqtrd 2315 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 1st `  ( 1st `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )  =  A )
2311, 22syl5eq 2327 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( 1st  o.  1st ) `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )  =  A )
245, 23eqtrd 2315 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( dom SetCat `  U
) `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )  =  A )
252, 24eqtrd 2315 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( dom SetCat `  U
) `  ( ( Id SetCat `  U ) `  A ) )  =  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    o. ccom 4693   Fun wfun 5249   -->wf 5251   -onto->wfo 5253   ` cfv 5255   1stc1st 6120   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   Id SetCatcidcase 25939
This theorem is referenced by:  domidmor2  25949  cmpidmor2  25969  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-map 6774  df-morcatset 25911  df-domcatset 25920  df-idcatset 25940
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