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Theorem isrocatset 26060
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 5541 . . . . . 6  |-  ( u  =  U  ->  ( Morphism SetCat `  u )  =  (
Morphism
SetCat `  U ) )
21eleq2d 2363 . . . . 5  |-  ( u  =  U  ->  (
a  e.  ( Morphism SetCat `  u )  <->  a  e.  ( Morphism SetCat `  U )
) )
31eleq2d 2363 . . . . 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 5918 . 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 26059 . 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 5879 . . . 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 5555 . . . . 5  |-  ( Morphism SetCat `  U )  e.  _V
109, 9mpt2ex 6214 . . . 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 2367 . . 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 5952 . . 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 4175 . 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 5618 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    o. ccom 4709   ` cfv 5271   {coprab 5875    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Univcgru 8428   Morphism SetCatccmrcase 26013   ro SetCatcrocase 26058
This theorem is referenced by:  cmp2morp  26061  cmpmorfun  26074  cmppar2  26075
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-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-rocatset 26059
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