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Theorem domidmor 26051
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 26049 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( Id SetCat `  U
) `  A )  =  <. <. A ,  A >. ,  (  _I  |`  A )
>. )
21fveq2d 5545 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( dom SetCat `  U
) `  ( ( Id SetCat `  U ) `  A ) )  =  ( ( dom SetCat `  U
) `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )
)
3 idmorimor 26050 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  <. <. A ,  A >. ,  (  _I  |`  A ) >.  e.  (
Morphism
SetCat `  U ) )
4 domcatval 26033 . . . 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 6155 . . . . . 6  |-  1st : _V -onto-> _V
7 fof 5467 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
86, 7ax-mp 8 . . . . 5  |-  1st : _V
--> _V
9 opex 4253 . . . . 5  |-  <. <. A ,  A >. ,  (  _I  |`  A ) >.  e.  _V
10 fvco3 5612 . . . . 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 4253 . . . . . . 7  |-  <. A ,  A >.  e.  _V
13 funi 5300 . . . . . . . 8  |-  Fun  _I
14 simpr 447 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  e.  U )
15 resfunexg 5753 . . . . . . . 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 6148 . . . . . . 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 5545 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 1st `  ( 1st `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )  =  ( 1st `  <. A ,  A >. ) )
20 op1stg 6148 . . . . . 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 2328 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 1st `  ( 1st `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )  =  A )
2311, 22syl5eq 2340 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( 1st  o.  1st ) `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )  =  A )
245, 23eqtrd 2328 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( dom SetCat `  U
) `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )  =  A )
252, 24eqtrd 2328 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    _I cid 4320    |` cres 4707    o. ccom 4709   Fun wfun 5265   -->wf 5267   -onto->wfo 5269   ` cfv 5271   1stc1st 6136   Univcgru 8428   Morphism SetCatccmrcase 26013   dom
SetCatcdomcase 26022   Id SetCatcidcase 26042
This theorem is referenced by:  domidmor2  26052  cmpidmor2  26072  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-map 6790  df-morcatset 26014  df-domcatset 26023  df-idcatset 26043
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