Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cmp2morpdom Unicode version

Theorem cmp2morpdom 26067
Description: Domain of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
cmp2morpcatt.1  |-  O  =  ( ro SetCat `  U
)
cmp2morpcatt.2  |- .Morphism  =  ( Morphism SetCat `  U )
cmp2morpcatt.3  |- .dom  =  ( dom SetCat `  U
)
cmp2morpcatt.4  |- .cod  =  ( cod SetCat `  U
)
Assertion
Ref Expression
cmp2morpdom  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.dom  `  ( A O B ) )  =  (.dom  `  B ) )

Proof of Theorem cmp2morpdom
StepHypRef Expression
1 cmp2morpcatt.3 . . . 4  |- .dom  =  ( dom SetCat `  U
)
21a1i 10 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  -> .dom  =  ( dom SetCat `  U )
)
32fveq1d 5543 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.dom  `  ( A O B ) )  =  ( ( dom SetCat `  U
) `  ( A O B ) ) )
4 simp1 955 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  U  e.  Univ )
5 cmp2morpcatt.2 . . . . . . 7  |- .Morphism  =  ( Morphism SetCat `  U )
65eleq2i 2360 . . . . . 6  |-  ( A  e. .Morphism  <->  A  e.  ( Morphism SetCat `  U )
)
76biimpi 186 . . . . 5  |-  ( A  e. .Morphism  ->  A  e.  ( Morphism SetCat `  U )
)
85eleq2i 2360 . . . . . 6  |-  ( B  e. .Morphism  <->  B  e.  ( Morphism SetCat `  U )
)
98biimpi 186 . . . . 5  |-  ( B  e. .Morphism  ->  B  e.  ( Morphism SetCat `  U )
)
107, 9anim12i 549 . . . 4  |-  ( ( A  e. .Morphism  /\  B  e. .Morphism  )  ->  ( A  e.  (
Morphism
SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) ) )
11 cmp2morpcatt.1 . . . . 5  |-  O  =  ( ro SetCat `  U
)
12 eqid 2296 . . . . 5  |-  ( Morphism SetCat `  U )  =  (
Morphism
SetCat `  U )
13 cmp2morpcatt.4 . . . . 5  |- .cod  =  ( cod SetCat `  U
)
1411, 12, 1, 13rocatval2 26063 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e.  (
Morphism
SetCat `  U ) )
1510, 14syl3an2 1216 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e.  (
Morphism
SetCat `  U ) )
16 domcatval 26033 . . 3  |-  ( ( U  e.  Univ  /\  ( A O B )  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  ( A O B ) )  =  ( ( 1st  o.  1st ) `  ( A O B ) ) )
174, 15, 16syl2anc 642 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( ( dom
SetCat `  U ) `  ( A O B ) )  =  ( ( 1st  o.  1st ) `  ( A O B ) ) )
1811, 5, 1, 13cmp2morpcats 26064 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
)
1918fveq2d 5545 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( ( 1st  o.  1st ) `  ( A O B ) )  =  ( ( 1st  o.  1st ) `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
)
20 fo1st 6155 . . . . . 6  |-  1st : _V -onto-> _V
21 fof 5467 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
2220, 21ax-mp 8 . . . . 5  |-  1st : _V
--> _V
23 opex 4253 . . . . 5  |-  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >.  e.  _V
24 fvco3 5612 . . . . 5  |-  ( ( 1st : _V --> _V  /\  <. <. (.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.  e.  _V )  ->  (
( 1st  o.  1st ) `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )  =  ( 1st `  ( 1st `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
) )
2522, 23, 24mp2an 653 . . . 4  |-  ( ( 1st  o.  1st ) `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )  =  ( 1st `  ( 1st `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
)
26 opex 4253 . . . . . 6  |-  <. (.dom  `  B ) ,  (.cod  `  A )
>.  e.  _V
27 fvex 5555 . . . . . . 7  |-  ( 2nd `  A )  e.  _V
28 fvex 5555 . . . . . . 7  |-  ( 2nd `  B )  e.  _V
2927, 28coex 5232 . . . . . 6  |-  ( ( 2nd `  A )  o.  ( 2nd `  B
) )  e.  _V
3026, 29op1st 6144 . . . . 5  |-  ( 1st `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )  =  <. (.dom  `  B ) ,  (.cod  `  A )
>.
3130fveq2i 5544 . . . 4  |-  ( 1st `  ( 1st `  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
) )  =  ( 1st `  <. (.dom  `  B ) ,  (.cod  `  A )
>. )
32 fvex 5555 . . . . 5  |-  (.dom  `  B )  e.  _V
33 fvex 5555 . . . . 5  |-  (.cod  `  A )  e.  _V
3432, 33op1st 6144 . . . 4  |-  ( 1st `  <. (.dom  `  B ) ,  (.cod  `  A )
>. )  =  (.dom  `  B )
3525, 31, 343eqtri 2320 . . 3  |-  ( ( 1st  o.  1st ) `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )  =  (.dom  `  B
)
3619, 35syl6eq 2344 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( ( 1st  o.  1st ) `  ( A O B ) )  =  (.dom  `  B ) )
373, 17, 363eqtrd 2332 1  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.dom  `  ( A O B ) )  =  (.dom  `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    o. ccom 4709   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Univcgru 8428   Morphism SetCatccmrcase 26013   dom
SetCatcdomcase 26022   cod
SetCatccodcase 26035   ro SetCatcrocase 26058
This theorem is referenced by:  cmpmorass  26069  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-2nd 6139  df-map 6790  df-morcatset 26014  df-domcatset 26023  df-codcatset 26036  df-rocatset 26059
  Copyright terms: Public domain W3C validator