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Theorem domcatfun 25925
Description: The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
domcatfun  |-  ( U  e.  Univ  ->  ( dom SetCat `
 U ) : ( Morphism SetCat `  U ) --> U )

Proof of Theorem domcatfun
Dummy variables  a  u  v  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-morcatset 25911 . . . . . . . 8  |-  Morphism SetCat  =  ( x  e.  Univ  |->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u ) ) } )
21a1i 10 . . . . . . 7  |-  ( U  e.  Univ  ->  Morphism SetCat  =  ( x  e.  Univ  |->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u ) ) } ) )
32fveq1d 5527 . . . . . 6  |-  ( U  e.  Univ  ->  ( Morphism SetCat `  U )  =  ( ( x  e.  Univ  |->  { <. <. u ,  v
>. ,  w >.  |  ( u  e.  x  /\  v  e.  x  /\  w  e.  (
v  ^m  u )
) } ) `  U ) )
43eleq2d 2350 . . . . 5  |-  ( U  e.  Univ  ->  ( a  e.  ( Morphism SetCat `  U
)  <->  a  e.  ( ( x  e.  Univ  |->  { <. <. u ,  v
>. ,  w >.  |  ( u  e.  x  /\  v  e.  x  /\  w  e.  (
v  ^m  u )
) } ) `  U ) ) )
54biimpa 470 . . . 4  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )
)  ->  a  e.  ( ( x  e. 
Univ  |->  { <. <. u ,  v >. ,  w >.  |  ( u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u
) ) } ) `
 U ) )
6 prismorcsetlem 25912 . . . . . . . . . 10  |-  ( U  e.  Univ  ->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u ) ) }  e.  _V )
7 eleq2 2344 . . . . . . . . . . . . 13  |-  ( x  =  U  ->  (
u  e.  x  <->  u  e.  U ) )
8 eleq2 2344 . . . . . . . . . . . . 13  |-  ( x  =  U  ->  (
v  e.  x  <->  v  e.  U ) )
97, 83anbi12d 1253 . . . . . . . . . . . 12  |-  ( x  =  U  ->  (
( u  e.  x  /\  v  e.  x  /\  w  e.  (
v  ^m  u )
)  <->  ( u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u
) ) ) )
109oprabbidv 5902 . . . . . . . . . . 11  |-  ( x  =  U  ->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u ) ) }  =  { <. <. u ,  v >. ,  w >.  |  ( u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u
) ) } )
11 eqid 2283 . . . . . . . . . . 11  |-  ( x  e.  Univ  |->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u ) ) } )  =  ( x  e.  Univ  |->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u ) ) } )
1210, 11fvmptg 5600 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  { <. <. u ,  v
>. ,  w >.  |  ( u  e.  U  /\  v  e.  U  /\  w  e.  (
v  ^m  u )
) }  e.  _V )  ->  ( ( x  e.  Univ  |->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u ) ) } ) `  U )  =  { <. <. u ,  v >. ,  w >.  |  ( u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u
) ) } )
136, 12mpdan 649 . . . . . . . . 9  |-  ( U  e.  Univ  ->  ( ( x  e.  Univ  |->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u ) ) } ) `  U )  =  { <. <. u ,  v >. ,  w >.  |  ( u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u
) ) } )
1413eleq2d 2350 . . . . . . . 8  |-  ( U  e.  Univ  ->  ( a  e.  ( ( x  e.  Univ  |->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u ) ) } ) `  U )  <-> 
a  e.  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u ) ) } ) )
1514biimpd 198 . . . . . . 7  |-  ( U  e.  Univ  ->  ( a  e.  ( ( x  e.  Univ  |->  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u ) ) } ) `  U )  ->  a  e.  { <. <. u ,  v
>. ,  w >.  |  ( u  e.  U  /\  v  e.  U  /\  w  e.  (
v  ^m  u )
) } ) )
1615adantr 451 . . . . . 6  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )
)  ->  ( a  e.  ( ( x  e. 
Univ  |->  { <. <. u ,  v >. ,  w >.  |  ( u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u
) ) } ) `
 U )  -> 
a  e.  { <. <.
u ,  v >. ,  w >.  |  (
u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u ) ) } ) )
17 eleq1 2343 . . . . . . . . 9  |-  ( u  =  ( 1st `  ( 1st `  a ) )  ->  ( u  e.  U  <->  ( 1st `  ( 1st `  a ) )  e.  U ) )
18 oveq2 5866 . . . . . . . . . 10  |-  ( u  =  ( 1st `  ( 1st `  a ) )  ->  ( v  ^m  u )  =  ( v  ^m  ( 1st `  ( 1st `  a
) ) ) )
1918eleq2d 2350 . . . . . . . . 9  |-  ( u  =  ( 1st `  ( 1st `  a ) )  ->  ( w  e.  ( v  ^m  u
)  <->  w  e.  (
v  ^m  ( 1st `  ( 1st `  a
) ) ) ) )
2017, 193anbi13d 1254 . . . . . . . 8  |-  ( u  =  ( 1st `  ( 1st `  a ) )  ->  ( ( u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u
) )  <->  ( ( 1st `  ( 1st `  a
) )  e.  U  /\  v  e.  U  /\  w  e.  (
v  ^m  ( 1st `  ( 1st `  a
) ) ) ) ) )
21 eleq1 2343 . . . . . . . . 9  |-  ( v  =  ( 2nd `  ( 1st `  a ) )  ->  ( v  e.  U  <->  ( 2nd `  ( 1st `  a ) )  e.  U ) )
22 oveq1 5865 . . . . . . . . . 10  |-  ( v  =  ( 2nd `  ( 1st `  a ) )  ->  ( v  ^m  ( 1st `  ( 1st `  a ) ) )  =  ( ( 2nd `  ( 1st `  a
) )  ^m  ( 1st `  ( 1st `  a
) ) ) )
2322eleq2d 2350 . . . . . . . . 9  |-  ( v  =  ( 2nd `  ( 1st `  a ) )  ->  ( w  e.  ( v  ^m  ( 1st `  ( 1st `  a
) ) )  <->  w  e.  ( ( 2nd `  ( 1st `  a ) )  ^m  ( 1st `  ( 1st `  a ) ) ) ) )
2421, 233anbi23d 1255 . . . . . . . 8  |-  ( v  =  ( 2nd `  ( 1st `  a ) )  ->  ( ( ( 1st `  ( 1st `  a ) )  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  ( 1st `  ( 1st `  a
) ) ) )  <-> 
( ( 1st `  ( 1st `  a ) )  e.  U  /\  ( 2nd `  ( 1st `  a
) )  e.  U  /\  w  e.  (
( 2nd `  ( 1st `  a ) )  ^m  ( 1st `  ( 1st `  a ) ) ) ) ) )
25 eleq1 2343 . . . . . . . . 9  |-  ( w  =  ( 2nd `  a
)  ->  ( w  e.  ( ( 2nd `  ( 1st `  a ) )  ^m  ( 1st `  ( 1st `  a ) ) )  <->  ( 2nd `  a
)  e.  ( ( 2nd `  ( 1st `  a ) )  ^m  ( 1st `  ( 1st `  a ) ) ) ) )
26253anbi3d 1258 . . . . . . . 8  |-  ( w  =  ( 2nd `  a
)  ->  ( (
( 1st `  ( 1st `  a ) )  e.  U  /\  ( 2nd `  ( 1st `  a
) )  e.  U  /\  w  e.  (
( 2nd `  ( 1st `  a ) )  ^m  ( 1st `  ( 1st `  a ) ) ) )  <->  ( ( 1st `  ( 1st `  a
) )  e.  U  /\  ( 2nd `  ( 1st `  a ) )  e.  U  /\  ( 2nd `  a )  e.  ( ( 2nd `  ( 1st `  a ) )  ^m  ( 1st `  ( 1st `  a ) ) ) ) ) )
2720, 24, 26eloprabi 6186 . . . . . . 7  |-  ( a  e.  { <. <. u ,  v >. ,  w >.  |  ( u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u
) ) }  ->  ( ( 1st `  ( 1st `  a ) )  e.  U  /\  ( 2nd `  ( 1st `  a
) )  e.  U  /\  ( 2nd `  a
)  e.  ( ( 2nd `  ( 1st `  a ) )  ^m  ( 1st `  ( 1st `  a ) ) ) ) )
28 fo1st 6139 . . . . . . . . . . . . . 14  |-  1st : _V -onto-> _V
29 fof 5451 . . . . . . . . . . . . . 14  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
3028, 29ax-mp 8 . . . . . . . . . . . . 13  |-  1st : _V
--> _V
31 vex 2791 . . . . . . . . . . . . 13  |-  a  e. 
_V
32 fvco3 5596 . . . . . . . . . . . . 13  |-  ( ( 1st : _V --> _V  /\  a  e.  _V )  ->  ( ( 1st  o.  1st ) `  a )  =  ( 1st `  ( 1st `  a ) ) )
3330, 31, 32mp2an 653 . . . . . . . . . . . 12  |-  ( ( 1st  o.  1st ) `  a )  =  ( 1st `  ( 1st `  a ) )
3433eqcomi 2287 . . . . . . . . . . 11  |-  ( 1st `  ( 1st `  a
) )  =  ( ( 1st  o.  1st ) `  a )
3534eleq1i 2346 . . . . . . . . . 10  |-  ( ( 1st `  ( 1st `  a ) )  e.  U  <->  ( ( 1st 
o.  1st ) `  a
)  e.  U )
3635biimpi 186 . . . . . . . . 9  |-  ( ( 1st `  ( 1st `  a ) )  e.  U  ->  ( ( 1st  o.  1st ) `  a )  e.  U
)
37363ad2ant1 976 . . . . . . . 8  |-  ( ( ( 1st `  ( 1st `  a ) )  e.  U  /\  ( 2nd `  ( 1st `  a
) )  e.  U  /\  ( 2nd `  a
)  e.  ( ( 2nd `  ( 1st `  a ) )  ^m  ( 1st `  ( 1st `  a ) ) ) )  ->  ( ( 1st  o.  1st ) `  a )  e.  U
)
3837a1d 22 . . . . . . 7  |-  ( ( ( 1st `  ( 1st `  a ) )  e.  U  /\  ( 2nd `  ( 1st `  a
) )  e.  U  /\  ( 2nd `  a
)  e.  ( ( 2nd `  ( 1st `  a ) )  ^m  ( 1st `  ( 1st `  a ) ) ) )  ->  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )
)  ->  ( ( 1st  o.  1st ) `  a )  e.  U
) )
3927, 38syl 15 . . . . . 6  |-  ( a  e.  { <. <. u ,  v >. ,  w >.  |  ( u  e.  U  /\  v  e.  U  /\  w  e.  ( v  ^m  u
) ) }  ->  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U ) )  -> 
( ( 1st  o.  1st ) `  a )  e.  U ) )
4016, 39syl6 29 . . . . 5  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )
)  ->  ( a  e.  ( ( x  e. 
Univ  |->  { <. <. u ,  v >. ,  w >.  |  ( u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u
) ) } ) `
 U )  -> 
( ( U  e. 
Univ  /\  a  e.  (
Morphism
SetCat `  U ) )  ->  ( ( 1st 
o.  1st ) `  a
)  e.  U ) ) )
4140pm2.43a 45 . . . 4  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )
)  ->  ( a  e.  ( ( x  e. 
Univ  |->  { <. <. u ,  v >. ,  w >.  |  ( u  e.  x  /\  v  e.  x  /\  w  e.  ( v  ^m  u
) ) } ) `
 U )  -> 
( ( 1st  o.  1st ) `  a )  e.  U ) )
425, 41mpd 14 . . 3  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )
)  ->  ( ( 1st  o.  1st ) `  a )  e.  U
)
43 eqid 2283 . . 3  |-  ( a  e.  ( Morphism SetCat `  U
)  |->  ( ( 1st 
o.  1st ) `  a
) )  =  ( a  e.  ( Morphism SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a ) )
4442, 43fmptd 5684 . 2  |-  ( U  e.  Univ  ->  ( a  e.  ( Morphism SetCat `  U
)  |->  ( ( 1st 
o.  1st ) `  a
) ) : (
Morphism
SetCat `  U ) --> U )
45 fveq2 5525 . . . . 5  |-  ( x  =  U  ->  ( Morphism SetCat `  x )  =  (
Morphism
SetCat `  U ) )
46 eqidd 2284 . . . . 5  |-  ( x  =  U  ->  (
( 1st  o.  1st ) `  a )  =  ( ( 1st 
o.  1st ) `  a
) )
4745, 46mpteq12dv 4098 . . . 4  |-  ( x  =  U  ->  (
a  e.  ( Morphism SetCat `  x )  |->  ( ( 1st  o.  1st ) `  a ) )  =  ( a  e.  (
Morphism
SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a )
) )
48 df-domcatset 25920 . . . 4  |-  dom SetCat  =  ( x  e.  Univ  |->  ( a  e.  ( Morphism SetCat `  x
)  |->  ( ( 1st 
o.  1st ) `  a
) ) )
49 fvex 5539 . . . . 5  |-  ( Morphism SetCat `  U )  e.  _V
5049mptex 5746 . . . 4  |-  ( a  e.  ( Morphism SetCat `  U
)  |->  ( ( 1st 
o.  1st ) `  a
) )  e.  _V
5147, 48, 50fvmpt 5602 . . 3  |-  ( U  e.  Univ  ->  ( dom SetCat `
 U )  =  ( a  e.  (
Morphism
SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a )
) )
5251feq1d 5379 . 2  |-  ( U  e.  Univ  ->  ( ( dom SetCat `  U ) : ( Morphism SetCat `  U
) --> U  <->  ( a  e.  ( Morphism SetCat `  U )  |->  ( ( 1st  o.  1st ) `  a ) ) : ( Morphism SetCat `  U ) --> U ) )
5344, 52mpbird 223 1  |-  ( U  e.  Univ  ->  ( dom SetCat `
 U ) : ( Morphism SetCat `  U ) --> U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   {coprab 5859   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919
This theorem is referenced by:  domdomcatfun  25926  domcatsetval  25928  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-1st 6122  df-2nd 6123  df-morcatset 25911  df-domcatset 25920
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