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Theorem iscatset 25978
Description: The category Set. (Contributed by FL, 8-Nov-2013.)
Assertion
Ref Expression
iscatset  |-  ( U  e.  Univ  ->  ( SetCat OLD `  U )  =  <. <.
( dom SetCat `  U
) ,  ( cod SetCat `
 U ) >. ,  <. ( Id SetCat `  U ) ,  ( ro SetCat `  U ) >. >. )

Proof of Theorem iscatset
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( x  =  U  ->  ( dom
SetCat `  x )  =  ( dom SetCat `  U
) )
2 fveq2 5525 . . . 4  |-  ( x  =  U  ->  ( cod
SetCat `  x )  =  ( cod SetCat `  U
) )
31, 2opeq12d 3804 . . 3  |-  ( x  =  U  ->  <. ( dom
SetCat `  x ) ,  ( cod SetCat `  x
) >.  =  <. ( dom
SetCat `  U ) ,  ( cod SetCat `  U
) >. )
4 fveq2 5525 . . . 4  |-  ( x  =  U  ->  ( Id SetCat `  x )  =  ( Id SetCat `  U ) )
5 fveq2 5525 . . . 4  |-  ( x  =  U  ->  ( ro SetCat `  x )  =  ( ro SetCat `  U ) )
64, 5opeq12d 3804 . . 3  |-  ( x  =  U  ->  <. ( Id SetCat `  x ) ,  ( ro SetCat `  x ) >.  =  <. ( Id SetCat `  U ) ,  ( ro SetCat `  U ) >. )
73, 6opeq12d 3804 . 2  |-  ( x  =  U  ->  <. <. ( dom
SetCat `  x ) ,  ( cod SetCat `  x
) >. ,  <. ( Id SetCat `  x ) ,  ( ro SetCat `  x ) >. >.  =  <. <.
( dom SetCat `  U
) ,  ( cod SetCat `
 U ) >. ,  <. ( Id SetCat `  U ) ,  ( ro SetCat `  U ) >. >. )
8 df-catset 25977 . 2  |-  SetCat OLD  =  ( x  e.  Univ  |->  <. <. ( dom SetCat `  x
) ,  ( cod SetCat `
 x ) >. ,  <. ( Id SetCat `  x ) ,  ( ro SetCat `  x ) >. >. )
9 opex 4237 . 2  |-  <. <. ( dom
SetCat `  U ) ,  ( cod SetCat `  U
) >. ,  <. ( Id SetCat `  U ) ,  ( ro SetCat `  U ) >. >.  e.  _V
107, 8, 9fvmpt 5602 1  |-  ( U  e.  Univ  ->  ( SetCat OLD `  U )  =  <. <.
( dom SetCat `  U
) ,  ( cod SetCat `
 U ) >. ,  <. ( Id SetCat `  U ) ,  ( ro SetCat `  U ) >. >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   <.cop 3643   ` cfv 5255   Univcgru 8412   dom SetCatcdomcase 25919   cod
SetCatccodcase 25932   Id SetCatcidcase 25939   ro SetCatcrocase 25955   SetCat OLDccaset 25976
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-iota 5219  df-fun 5257  df-fv 5263  df-catset 25977
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