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Theorem rocatval 25959
Description: The composite of two morphisms in the category Set is a morphism. (Contributed by FL, 6-Nov-2013.)
Hypothesis
Ref Expression
cmp2morp.1  |-  O  =  ( ro SetCat `  U
)
Assertion
Ref Expression
rocatval  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( A O B )  e.  (
Morphism
SetCat `  U ) )

Proof of Theorem rocatval
StepHypRef Expression
1 cmp2morp.1 . . 3  |-  O  =  ( ro SetCat `  U
)
21cmp2morp 25958 . 2  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( A O B )  =  <. <.
( ( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
3 simp1 955 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  U  e.  Univ )
4 simp2r 982 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  B  e.  ( Morphism SetCat `  U )
)
5 domcatval 25930 . . . . 5  |-  ( ( U  e.  Univ  /\  B  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  B )  =  ( ( 1st  o.  1st ) `  B )
)
63, 4, 5syl2anc 642 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( dom
SetCat `  U ) `  B )  =  ( ( 1st  o.  1st ) `  B )
)
7 domcatsetval 25928 . . . . 5  |-  ( ( U  e.  Univ  /\  B  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  B )  e.  U
)
83, 4, 7syl2anc 642 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( dom
SetCat `  U ) `  B )  e.  U
)
96, 8eqeltrrd 2358 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( 1st  o.  1st ) `  B )  e.  U
)
10 simp2l 981 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  A  e.  ( Morphism SetCat `  U )
)
11 codcatval 25936 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( cod
SetCat `  U ) `  A )  =  ( ( 2nd  o.  1st ) `  A )
)
123, 10, 11syl2anc 642 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( cod
SetCat `  U ) `  A )  =  ( ( 2nd  o.  1st ) `  A )
)
13 codcatsetval 25935 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( cod
SetCat `  U ) `  A )  e.  U
)
143, 10, 13syl2anc 642 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( cod
SetCat `  U ) `  A )  e.  U
)
1512, 14eqeltrrd 2358 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( 2nd  o.  1st ) `  A )  e.  U
)
16 eqid 2283 . . . . . . . . . 10  |-  ( ( 1st  o.  1st ) `  A )  =  ( ( 1st  o.  1st ) `  A )
17 eqid 2283 . . . . . . . . . 10  |-  ( ( 2nd  o.  1st ) `  A )  =  ( ( 2nd  o.  1st ) `  A )
18 eqid 2283 . . . . . . . . . 10  |-  ( 2nd `  A )  =  ( 2nd `  A )
1916, 17, 18prismorcset2 25918 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( (
( 1st  o.  1st ) `  A )  e.  U  /\  (
( 2nd  o.  1st ) `  A )  e.  U  /\  ( 2nd `  A )  e.  ( ( ( 2nd 
o.  1st ) `  A
)  ^m  ( ( 1st  o.  1st ) `  A ) ) ) )
203, 10, 19syl2anc 642 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( (
( 1st  o.  1st ) `  A )  e.  U  /\  (
( 2nd  o.  1st ) `  A )  e.  U  /\  ( 2nd `  A )  e.  ( ( ( 2nd 
o.  1st ) `  A
)  ^m  ( ( 1st  o.  1st ) `  A ) ) ) )
2120simp3d 969 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( 2nd `  A )  e.  ( ( ( 2nd  o.  1st ) `  A )  ^m  ( ( 1st 
o.  1st ) `  A
) ) )
22 domcatval 25930 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  A )  =  ( ( 1st  o.  1st ) `  A )
)
2322adantrr 697 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
) )  ->  (
( dom SetCat `  U
) `  A )  =  ( ( 1st 
o.  1st ) `  A
) )
24 codcatval 25936 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  B  e.  ( Morphism SetCat `  U )
)  ->  ( ( cod
SetCat `  U ) `  B )  =  ( ( 2nd  o.  1st ) `  B )
)
2524adantrl 696 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
) )  ->  (
( cod SetCat `  U
) `  B )  =  ( ( 2nd 
o.  1st ) `  B
) )
2623, 25eqeq12d 2297 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
) )  ->  (
( ( dom SetCat `  U
) `  A )  =  ( ( cod SetCat `
 U ) `  B )  <->  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd  o.  1st ) `  B )
) )
2726biimp3a 1281 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd  o.  1st ) `  B )
)
2827eqcomd 2288 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( 2nd  o.  1st ) `  B )  =  ( ( 1st  o.  1st ) `  A )
)
2928oveq2d 5874 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( (
( 2nd  o.  1st ) `  A )  ^m  ( ( 2nd  o.  1st ) `  B ) )  =  ( ( ( 2nd  o.  1st ) `  A )  ^m  ( ( 1st  o.  1st ) `  A ) ) )
3021, 29eleqtrrd 2360 . . . . . 6  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( 2nd `  A )  e.  ( ( ( 2nd  o.  1st ) `  A )  ^m  ( ( 2nd 
o.  1st ) `  B
) ) )
31 fvex 5539 . . . . . . 7  |-  ( ( 2nd  o.  1st ) `  A )  e.  _V
32 fvex 5539 . . . . . . 7  |-  ( ( 2nd  o.  1st ) `  B )  e.  _V
3331, 32elmap 6796 . . . . . 6  |-  ( ( 2nd `  A )  e.  ( ( ( 2nd  o.  1st ) `  A )  ^m  (
( 2nd  o.  1st ) `  B )
)  <->  ( 2nd `  A
) : ( ( 2nd  o.  1st ) `  B ) --> ( ( 2nd  o.  1st ) `  A ) )
3430, 33sylib 188 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( 2nd `  A ) : ( ( 2nd  o.  1st ) `  B ) --> ( ( 2nd  o.  1st ) `  A ) )
35 eqid 2283 . . . . . . . . 9  |-  ( ( 1st  o.  1st ) `  B )  =  ( ( 1st  o.  1st ) `  B )
36 eqid 2283 . . . . . . . . 9  |-  ( ( 2nd  o.  1st ) `  B )  =  ( ( 2nd  o.  1st ) `  B )
37 eqid 2283 . . . . . . . . 9  |-  ( 2nd `  B )  =  ( 2nd `  B )
3835, 36, 37prismorcset2 25918 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  B  e.  ( Morphism SetCat `  U )
)  ->  ( (
( 1st  o.  1st ) `  B )  e.  U  /\  (
( 2nd  o.  1st ) `  B )  e.  U  /\  ( 2nd `  B )  e.  ( ( ( 2nd 
o.  1st ) `  B
)  ^m  ( ( 1st  o.  1st ) `  B ) ) ) )
393, 4, 38syl2anc 642 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( (
( 1st  o.  1st ) `  B )  e.  U  /\  (
( 2nd  o.  1st ) `  B )  e.  U  /\  ( 2nd `  B )  e.  ( ( ( 2nd 
o.  1st ) `  B
)  ^m  ( ( 1st  o.  1st ) `  B ) ) ) )
4039simp3d 969 . . . . . 6  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( 2nd `  B )  e.  ( ( ( 2nd  o.  1st ) `  B )  ^m  ( ( 1st 
o.  1st ) `  B
) ) )
41 fvex 5539 . . . . . . 7  |-  ( ( 1st  o.  1st ) `  B )  e.  _V
4232, 41elmap 6796 . . . . . 6  |-  ( ( 2nd `  B )  e.  ( ( ( 2nd  o.  1st ) `  B )  ^m  (
( 1st  o.  1st ) `  B )
)  <->  ( 2nd `  B
) : ( ( 1st  o.  1st ) `  B ) --> ( ( 2nd  o.  1st ) `  B ) )
4340, 42sylib 188 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( 2nd `  B ) : ( ( 1st  o.  1st ) `  B ) --> ( ( 2nd  o.  1st ) `  B ) )
44 fco 5398 . . . . 5  |-  ( ( ( 2nd `  A
) : ( ( 2nd  o.  1st ) `  B ) --> ( ( 2nd  o.  1st ) `  A )  /\  ( 2nd `  B ) : ( ( 1st  o.  1st ) `  B ) --> ( ( 2nd  o.  1st ) `  B ) )  ->  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) : ( ( 1st  o.  1st ) `  B ) --> ( ( 2nd  o.  1st ) `  A ) )
4534, 43, 44syl2anc 642 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) : ( ( 1st  o.  1st ) `  B ) --> ( ( 2nd  o.  1st ) `  A ) )
4631, 41elmap 6796 . . . 4  |-  ( ( ( 2nd `  A
)  o.  ( 2nd `  B ) )  e.  ( ( ( 2nd 
o.  1st ) `  A
)  ^m  ( ( 1st  o.  1st ) `  B ) )  <->  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) : ( ( 1st  o.  1st ) `  B ) --> ( ( 2nd  o.  1st ) `  A ) )
4745, 46sylibr 203 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( 2nd `  A )  o.  ( 2nd `  B
) )  e.  ( ( ( 2nd  o.  1st ) `  A )  ^m  ( ( 1st 
o.  1st ) `  B
) ) )
4841a1i 10 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( 1st  o.  1st ) `  B )  e.  _V )
4931a1i 10 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( 2nd  o.  1st ) `  A )  e.  _V )
50 fvex 5539 . . . . . 6  |-  ( 2nd `  A )  e.  _V
51 fvex 5539 . . . . . 6  |-  ( 2nd `  B )  e.  _V
5250, 51coex 5216 . . . . 5  |-  ( ( 2nd `  A )  o.  ( 2nd `  B
) )  e.  _V
5352a1i 10 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( ( 2nd `  A )  o.  ( 2nd `  B
) )  e.  _V )
54 prismorcset 25914 . . . 4  |-  ( ( ( ( ( 1st 
o.  1st ) `  B
)  e.  _V  /\  ( ( 2nd  o.  1st ) `  A )  e.  _V  /\  (
( 2nd `  A
)  o.  ( 2nd `  B ) )  e. 
_V )  /\  U  e.  Univ )  ->  ( <. <. ( ( 1st 
o.  1st ) `  B
) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.  e.  ( Morphism SetCat `  U )  <->  ( ( ( 1st  o.  1st ) `  B )  e.  U  /\  (
( 2nd  o.  1st ) `  A )  e.  U  /\  (
( 2nd `  A
)  o.  ( 2nd `  B ) )  e.  ( ( ( 2nd 
o.  1st ) `  A
)  ^m  ( ( 1st  o.  1st ) `  B ) ) ) ) )
5548, 49, 53, 3, 54syl31anc 1185 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( <. <.
( ( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >.  e.  (
Morphism
SetCat `  U )  <->  ( (
( 1st  o.  1st ) `  B )  e.  U  /\  (
( 2nd  o.  1st ) `  A )  e.  U  /\  (
( 2nd `  A
)  o.  ( 2nd `  B ) )  e.  ( ( ( 2nd 
o.  1st ) `  A
)  ^m  ( ( 1st  o.  1st ) `  B ) ) ) ) )
569, 15, 47, 55mpbir3and 1135 . 2  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  <. <. (
( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >.  e.  (
Morphism
SetCat `  U ) )
572, 56eqeltrd 2357 1  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( A O B )  e.  (
Morphism
SetCat `  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   cod
SetCatccodcase 25932   ro SetCatcrocase 25955
This theorem is referenced by:  rocatval2  25960  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-morcatset 25911  df-domcatset 25920  df-codcatset 25933  df-rocatset 25956
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