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Theorem domcatval 25930
Description: The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
domcatval  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  A )  =  ( ( 1st  o.  1st ) `  A )
)

Proof of Theorem domcatval
Dummy variables  a  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  U  e.  Univ )
2 fvex 5539 . . . . 5  |-  ( Morphism SetCat `  U )  e.  _V
32mptex 5746 . . . 4  |-  ( a  e.  ( Morphism SetCat `  U
)  |->  ( ( 1st 
o.  1st ) `  a
) )  e.  _V
4 fveq2 5525 . . . . . 6  |-  ( u  =  U  ->  ( Morphism SetCat `  u )  =  (
Morphism
SetCat `  U ) )
5 eqidd 2284 . . . . . 6  |-  ( u  =  U  ->  (
( 1st  o.  1st ) `  a )  =  ( ( 1st 
o.  1st ) `  a
) )
64, 5mpteq12dv 4098 . . . . 5  |-  ( u  =  U  ->  (
a  e.  ( Morphism SetCat `  u )  |->  ( ( 1st  o.  1st ) `  a ) )  =  ( a  e.  (
Morphism
SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a )
) )
7 df-domcatset 25920 . . . . 5  |-  dom SetCat  =  ( u  e.  Univ  |->  ( a  e.  ( Morphism SetCat `  u
)  |->  ( ( 1st 
o.  1st ) `  a
) ) )
86, 7fvmptg 5600 . . . 4  |-  ( ( U  e.  Univ  /\  (
a  e.  ( Morphism SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a ) )  e. 
_V )  ->  ( dom
SetCat `  U )  =  ( a  e.  (
Morphism
SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a )
) )
91, 3, 8sylancl 643 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( dom SetCat `  U )  =  ( a  e.  ( Morphism SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a ) ) )
109fveq1d 5527 . 2  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  A )  =  ( ( a  e.  (
Morphism
SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a )
) `  A )
)
11 simpr 447 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  A  e.  ( Morphism SetCat `  U )
)
12 fvex 5539 . . 3  |-  ( ( 1st  o.  1st ) `  A )  e.  _V
13 fveq2 5525 . . . 4  |-  ( a  =  A  ->  (
( 1st  o.  1st ) `  a )  =  ( ( 1st 
o.  1st ) `  A
) )
14 eqid 2283 . . . 4  |-  ( a  e.  ( Morphism SetCat `  U
)  |->  ( ( 1st 
o.  1st ) `  a
) )  =  ( a  e.  ( Morphism SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a ) )
1513, 14fvmptg 5600 . . 3  |-  ( ( A  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  e.  _V )  ->  (
( a  e.  (
Morphism
SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a )
) `  A )  =  ( ( 1st 
o.  1st ) `  A
) )
1611, 12, 15sylancl 643 . 2  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( (
a  e.  ( Morphism SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a ) ) `  A )  =  ( ( 1st  o.  1st ) `  A )
)
1710, 16eqtrd 2315 1  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  A )  =  ( ( 1st  o.  1st ) `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077    o. ccom 4693   ` cfv 5255   1stc1st 6120   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919
This theorem is referenced by:  domcatval2  25931  prismorcset3  25938  domidmor  25948  cmp2morp  25958  rocatval  25959  cmp2morpcats  25961  cmp2morpdom  25964  cmppar3  25974  cmpmor  25975
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-rep 4131  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-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-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-domcatset 25920
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