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Theorem cmppar2 25972
Description: Morphisms composition is defined every time the codomain of the second operand matches the domain of the first one. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
Assertion
Ref Expression
cmppar2  |-  ( 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
) ) } )
Distinct variable group:    U, a, b

Proof of Theorem cmppar2
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 isrocatset 25957 . . . 4  |-  ( U  e.  Univ  ->  ( ro SetCat `
 U )  =  { <. <. a ,  b
>. ,  c >.  |  ( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) } )
21dmeqd 4881 . . 3  |-  ( U  e.  Univ  ->  dom  ( ro SetCat `  U )  =  dom  { <. <. a ,  b >. ,  c
>.  |  ( (
a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) } )
3 dmoprab 5928 . . 3  |-  dom  { <. <. a ,  b
>. ,  c >.  |  ( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) }  =  { <. a ,  b >.  |  E. c ( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) }
42, 3syl6eq 2331 . 2  |-  ( U  e.  Univ  ->  dom  ( ro SetCat `  U )  =  { <. a ,  b
>.  |  E. c
( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) } )
5 opex 4237 . . . . 5  |-  <. <. (
( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >.  e.  _V
65isseti 2794 . . . 4  |-  E. c 
c  =  <. <. (
( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >.
7 19.42v 1846 . . . 4  |-  ( E. c ( ( a  e.  ( Morphism SetCat `  U
)  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
)  <->  ( ( a  e.  ( Morphism SetCat `  U
)  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  E. c  c  =  <. <.
( ( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >. )
)
86, 7mpbiran2 885 . . 3  |-  ( E. c ( ( a  e.  ( Morphism SetCat `  U
)  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
)  <->  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) )
98opabbii 4083 . 2  |-  { <. a ,  b >.  |  E. c ( ( a  e.  ( Morphism SetCat `  U
)  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) }  =  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }
104, 9syl6eq 2331 1  |-  ( 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
) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   <.cop 3643   {copab 4076   dom cdm 4689    o. ccom 4693   ` cfv 5255   {coprab 5859   1stc1st 6120   2ndc2nd 6121   Univcgru 8412   Morphism SetCatccmrcase 25910   ro SetCatcrocase 25955
This theorem is referenced by:  cmppar  25973  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-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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-rocatset 25956
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