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

Theorem cmpmor 26078
Description: The composite of two morphisms is a morphism. (Contributed by FL, 8-Nov-2013.)
Assertion
Ref Expression
cmpmor  |-  ( U  e.  Univ  ->  ran  ( ro SetCat `  U )  C_  ( Morphism SetCat `  U )
)

Proof of Theorem cmpmor
Dummy variables  x  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmpmorfun 26074 . . 3  |-  ( U  e.  Univ  ->  Fun  ( ro SetCat `  U )
)
2 cmppar2 26075 . . 3  |-  ( U  e.  Univ  ->  dom  ( ro SetCat `  U )  =  { <. a ,  b
>.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) } )
3 df-fn 5274 . . 3  |-  ( ( ro SetCat `  U )  Fn  { <. a ,  b
>.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }  <->  ( Fun  ( ro SetCat `  U )  /\  dom  ( ro SetCat `  U )  =  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) } ) )
41, 2, 3sylanbrc 645 . 2  |-  ( U  e.  Univ  ->  ( ro SetCat `
 U )  Fn 
{ <. a ,  b
>.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) } )
5 elopab 4288 . . . . 5  |-  ( x  e.  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }  <->  E. a E. b ( x  = 
<. a ,  b >.  /\  ( a  e.  (
Morphism
SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) ) )
6 domcatval 26033 . . . . . . . . . . . . . 14  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  a )  =  ( ( 1st  o.  1st ) `  a )
)
763adant3 975 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( dom SetCat `  U
) `  a )  =  ( ( 1st 
o.  1st ) `  a
) )
8 codcatval 26039 . . . . . . . . . . . . . 14  |-  ( ( U  e.  Univ  /\  b  e.  ( Morphism SetCat `  U )
)  ->  ( ( cod
SetCat `  U ) `  b )  =  ( ( 2nd  o.  1st ) `  b )
)
983adant2 974 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( cod SetCat `  U
) `  b )  =  ( ( 2nd 
o.  1st ) `  b
) )
10 eqtr 2313 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( dom SetCat `  U
) `  a )  =  ( ( 1st 
o.  1st ) `  a
)  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd  o.  1st ) `  b )
)  ->  ( ( dom
SetCat `  U ) `  a )  =  ( ( 2nd  o.  1st ) `  b )
)
11 eqtr 2313 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( dom SetCat `  U
) `  a )  =  ( ( 2nd 
o.  1st ) `  b
)  /\  ( ( 2nd  o.  1st ) `  b )  =  ( ( cod SetCat `  U
) `  b )
)  ->  ( ( dom
SetCat `  U ) `  a )  =  ( ( cod SetCat `  U
) `  b )
)
12 df-ov 5877 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( a ( ro SetCat `  U
) b )  =  ( ( ro SetCat `  U ) `  <. a ,  b >. )
13 eqid 2296 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ro SetCat `
 U )  =  ( ro SetCat `  U
)
1413rocatval 26062 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( U  e.  Univ  /\  (
a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  a )  =  ( ( cod SetCat `  U
) `  b )
)  ->  ( a
( ro SetCat `  U
) b )  e.  ( Morphism SetCat `  U )
)
15143expa 1151 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( U  e.  Univ  /\  ( a  e.  (
Morphism
SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) ) )  /\  ( ( dom SetCat `
 U ) `  a )  =  ( ( cod SetCat `  U
) `  b )
)  ->  ( a
( ro SetCat `  U
) b )  e.  ( Morphism SetCat `  U )
)
1612, 15syl5eqelr 2381 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( U  e.  Univ  /\  ( a  e.  (
Morphism
SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) ) )  /\  ( ( dom SetCat `
 U ) `  a )  =  ( ( cod SetCat `  U
) `  b )
)  ->  ( ( ro SetCat `  U ) `  <. a ,  b
>. )  e.  ( Morphism SetCat `  U ) )
1716exp42 594 . . . . . . . . . . . . . . . . . . . . 21  |-  ( U  e.  Univ  ->  ( a  e.  ( Morphism SetCat `  U
)  ->  ( b  e.  ( Morphism SetCat `  U )  ->  ( ( ( dom SetCat `
 U ) `  a )  =  ( ( cod SetCat `  U
) `  b )  ->  ( ( ro SetCat `  U ) `  <. a ,  b >. )  e.  ( Morphism SetCat `  U )
) ) ) )
18173imp 1145 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ( dom SetCat `
 U ) `  a )  =  ( ( cod SetCat `  U
) `  b )  ->  ( ( ro SetCat `  U ) `  <. a ,  b >. )  e.  ( Morphism SetCat `  U )
) )
1911, 18syl5com 26 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( dom SetCat `  U
) `  a )  =  ( ( 2nd 
o.  1st ) `  b
)  /\  ( ( 2nd  o.  1st ) `  b )  =  ( ( cod SetCat `  U
) `  b )
)  ->  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ro SetCat `  U ) `  <. a ,  b >. )  e.  ( Morphism SetCat `  U )
) )
2019expcom 424 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd  o.  1st ) `  b )  =  ( ( cod SetCat `
 U ) `  b )  ->  (
( ( dom SetCat `  U
) `  a )  =  ( ( 2nd 
o.  1st ) `  b
)  ->  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ro SetCat `  U ) `  <. a ,  b >. )  e.  ( Morphism SetCat `  U )
) ) )
2120eqcoms 2299 . . . . . . . . . . . . . . . . 17  |-  ( ( ( cod SetCat `  U
) `  b )  =  ( ( 2nd 
o.  1st ) `  b
)  ->  ( (
( dom SetCat `  U
) `  a )  =  ( ( 2nd 
o.  1st ) `  b
)  ->  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ro SetCat `  U ) `  <. a ,  b >. )  e.  ( Morphism SetCat `  U )
) ) )
2221com3l 75 . . . . . . . . . . . . . . . 16  |-  ( ( ( dom SetCat `  U
) `  a )  =  ( ( 2nd 
o.  1st ) `  b
)  ->  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ( cod SetCat `
 U ) `  b )  =  ( ( 2nd  o.  1st ) `  b )  ->  ( ( ro SetCat `  U ) `  <. a ,  b >. )  e.  ( Morphism SetCat `  U )
) ) )
2310, 22syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( dom SetCat `  U
) `  a )  =  ( ( 1st 
o.  1st ) `  a
)  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd  o.  1st ) `  b )
)  ->  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ( cod SetCat `
 U ) `  b )  =  ( ( 2nd  o.  1st ) `  b )  ->  ( ( ro SetCat `  U ) `  <. a ,  b >. )  e.  ( Morphism SetCat `  U )
) ) )
2423ex 423 . . . . . . . . . . . . . 14  |-  ( ( ( dom SetCat `  U
) `  a )  =  ( ( 1st 
o.  1st ) `  a
)  ->  ( (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
)  ->  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ( cod SetCat `
 U ) `  b )  =  ( ( 2nd  o.  1st ) `  b )  ->  ( ( ro SetCat `  U ) `  <. a ,  b >. )  e.  ( Morphism SetCat `  U )
) ) ) )
2524com24 81 . . . . . . . . . . . . 13  |-  ( ( ( dom SetCat `  U
) `  a )  =  ( ( 1st 
o.  1st ) `  a
)  ->  ( (
( cod SetCat `  U
) `  b )  =  ( ( 2nd 
o.  1st ) `  b
)  ->  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ( 1st 
o.  1st ) `  a
)  =  ( ( 2nd  o.  1st ) `  b )  ->  (
( ro SetCat `  U
) `  <. a ,  b >. )  e.  (
Morphism
SetCat `  U ) ) ) ) )
267, 9, 25sylc 56 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( U  e. 
Univ  /\  a  e.  (
Morphism
SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ( 1st 
o.  1st ) `  a
)  =  ( ( 2nd  o.  1st ) `  b )  ->  (
( ro SetCat `  U
) `  <. a ,  b >. )  e.  (
Morphism
SetCat `  U ) ) ) )
2726pm2.43i 43 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  -> 
( ( ( 1st 
o.  1st ) `  a
)  =  ( ( 2nd  o.  1st ) `  b )  ->  (
( ro SetCat `  U
) `  <. a ,  b >. )  e.  (
Morphism
SetCat `  U ) ) )
28273exp 1150 . . . . . . . . . 10  |-  ( U  e.  Univ  ->  ( a  e.  ( Morphism SetCat `  U
)  ->  ( b  e.  ( Morphism SetCat `  U )  ->  ( ( ( 1st 
o.  1st ) `  a
)  =  ( ( 2nd  o.  1st ) `  b )  ->  (
( ro SetCat `  U
) `  <. a ,  b >. )  e.  (
Morphism
SetCat `  U ) ) ) ) )
2928com4l 78 . . . . . . . . 9  |-  ( a  e.  ( Morphism SetCat `  U
)  ->  ( b  e.  ( Morphism SetCat `  U )  ->  ( ( ( 1st 
o.  1st ) `  a
)  =  ( ( 2nd  o.  1st ) `  b )  ->  ( U  e.  Univ  ->  (
( ro SetCat `  U
) `  <. a ,  b >. )  e.  (
Morphism
SetCat `  U ) ) ) ) )
30293imp 1145 . . . . . . . 8  |-  ( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  ->  ( U  e.  Univ  ->  (
( ro SetCat `  U
) `  <. a ,  b >. )  e.  (
Morphism
SetCat `  U ) ) )
31 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  <. a ,  b
>.  ->  ( ( ro SetCat `
 U ) `  x )  =  ( ( ro SetCat `  U
) `  <. a ,  b >. ) )
3231eleq1d 2362 . . . . . . . . 9  |-  ( x  =  <. a ,  b
>.  ->  ( ( ( ro SetCat `  U ) `  x )  e.  (
Morphism
SetCat `  U )  <->  ( ( ro SetCat `  U ) `  <. a ,  b
>. )  e.  ( Morphism SetCat `  U ) ) )
3332imbi2d 307 . . . . . . . 8  |-  ( x  =  <. a ,  b
>.  ->  ( ( U  e.  Univ  ->  ( ( ro SetCat `  U ) `  x )  e.  (
Morphism
SetCat `  U ) )  <-> 
( U  e.  Univ  -> 
( ( ro SetCat `  U ) `  <. a ,  b >. )  e.  ( Morphism SetCat `  U )
) ) )
3430, 33syl5ibr 212 . . . . . . 7  |-  ( x  =  <. a ,  b
>.  ->  ( ( a  e.  ( Morphism SetCat `  U
)  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  ->  ( U  e.  Univ  ->  (
( ro SetCat `  U
) `  x )  e.  ( Morphism SetCat `  U )
) ) )
3534imp 418 . . . . . 6  |-  ( ( x  =  <. a ,  b >.  /\  (
a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) )  -> 
( U  e.  Univ  -> 
( ( ro SetCat `  U ) `  x
)  e.  ( Morphism SetCat `  U ) ) )
3635exlimivv 1625 . . . . 5  |-  ( E. a E. b ( x  =  <. a ,  b >.  /\  (
a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) )  -> 
( U  e.  Univ  -> 
( ( ro SetCat `  U ) `  x
)  e.  ( Morphism SetCat `  U ) ) )
375, 36sylbi 187 . . . 4  |-  ( x  e.  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }  ->  ( U  e.  Univ  ->  ( ( ro SetCat `  U
) `  x )  e.  ( Morphism SetCat `  U )
) )
3837com12 27 . . 3  |-  ( U  e.  Univ  ->  ( x  e.  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }  ->  ( ( ro SetCat `  U
) `  x )  e.  ( Morphism SetCat `  U )
) )
3938ralrimiv 2638 . 2  |-  ( U  e.  Univ  ->  A. x  e.  { <. a ,  b
>.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }  (
( ro SetCat `  U
) `  x )  e.  ( Morphism SetCat `  U )
)
40 fnfvrnss 5703 . 2  |-  ( ( ( ro SetCat `  U
)  Fn  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }  /\  A. x  e.  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }  (
( ro SetCat `  U
) `  x )  e.  ( Morphism SetCat `  U )
)  ->  ran  ( ro SetCat `
 U )  C_  ( Morphism SetCat `  U )
)
414, 39, 40syl2anc 642 1  |-  ( U  e.  Univ  ->  ran  ( ro SetCat `  U )  C_  ( Morphism SetCat `  U )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   <.cop 3656   {copab 4092   dom cdm 4705   ran crn 4706    o. ccom 4709   Fun wfun 5265    Fn wfn 5266   ` 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:  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