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Theorem isrocatset 25957
Description: Definition of the composition of two morphisms in the category Set . (Contributed by FL, 6-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
Assertion
Ref Expression
isrocatset  |-  ( 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 ) ) >.
) } )
Distinct variable group:    U, a, b, c

Proof of Theorem isrocatset
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . . 6  |-  ( u  =  U  ->  ( Morphism SetCat `  u )  =  (
Morphism
SetCat `  U ) )
21eleq2d 2350 . . . . 5  |-  ( u  =  U  ->  (
a  e.  ( Morphism SetCat `  u )  <->  a  e.  ( Morphism SetCat `  U )
) )
31eleq2d 2350 . . . . 5  |-  ( u  =  U  ->  (
b  e.  ( Morphism SetCat `  u )  <->  b  e.  ( Morphism SetCat `  U )
) )
42, 33anbi12d 1253 . . . 4  |-  ( u  =  U  ->  (
( a  e.  (
Morphism
SetCat `  u )  /\  b  e.  ( Morphism SetCat `  u )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  <->  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) ) )
54anbi1d 685 . . 3  |-  ( u  =  U  ->  (
( ( 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
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) ) )
65oprabbidv 5902 . 2  |-  ( u  =  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 ) ) >.
) }  =  { <. <. 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 ) ) >.
) } )
7 df-rocatset 25956 . 2  |-  ro SetCat  =  ( u  e.  Univ  |->  { <. <. 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 ) ) >.
) } )
8 df-mpt2 5863 . . . 4  |-  ( a  e.  ( Morphism SetCat `  U
) ,  b  e.  ( Morphism SetCat `  U )  |-> 
<. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
)  =  { <. <.
a ,  b >. ,  c >.  |  ( ( a  e.  (
Morphism
SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  /\  c  =  <. <. (
( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >. ) }
9 fvex 5539 . . . . 5  |-  ( Morphism SetCat `  U )  e.  _V
109, 9mpt2ex 6198 . . . 4  |-  ( a  e.  ( Morphism SetCat `  U
) ,  b  e.  ( Morphism SetCat `  U )  |-> 
<. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
)  e.  _V
118, 10eqeltrri 2354 . . 3  |-  { <. <.
a ,  b >. ,  c >.  |  ( ( a  e.  (
Morphism
SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  /\  c  =  <. <. (
( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >. ) }  e.  _V
12 3simpa 952 . . . . 5  |-  ( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  ->  (
a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )
) )
1312anim1i 551 . . . 4  |-  ( ( ( 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 )
)  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) )
1413ssoprab2i 5936 . . 3  |-  { <. <.
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 ) ) >.
) }  C_  { <. <.
a ,  b >. ,  c >.  |  ( ( a  e.  (
Morphism
SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U ) )  /\  c  =  <. <. (
( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >. ) }
1511, 14ssexi 4159 . 2  |-  { <. <.
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 ) ) >.
) }  e.  _V
166, 7, 15fvmpt 5602 1  |-  ( 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 ) ) >.
) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    o. ccom 4693   ` cfv 5255   {coprab 5859    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Univcgru 8412   Morphism SetCatccmrcase 25910   ro SetCatcrocase 25955
This theorem is referenced by:  cmp2morp  25958  cmpmorfun  25971  cmppar2  25972
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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