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Theorem cmp2morp 25958
Description: Composite of two morphisms. (Contributed by FL, 6-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
Hypothesis
Ref Expression
cmp2morp.1  |-  O  =  ( ro SetCat `  U
)
Assertion
Ref Expression
cmp2morp  |-  ( ( 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
) ) >. )

Proof of Theorem cmp2morp
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmp2morp.1 . . . . . 6  |-  O  =  ( ro SetCat `  U
)
2 isrocatset 25957 . . . . . 6  |-  ( 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 ) ) >.
) } )
31, 2syl5eq 2327 . . . . 5  |-  ( U  e.  Univ  ->  O  =  { <. <. 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 ) ) >.
) } )
43oveqd 5875 . . . 4  |-  ( U  e.  Univ  ->  ( A O B )  =  ( A { <. <.
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 ) ) >.
) } B ) )
54adantr 451 . . 3  |-  ( ( U  e.  Univ  /\  (
( A  e.  (
Morphism
SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  /\  ( ( dom SetCat `  U
) `  A )  =  ( ( cod SetCat `
 U ) `  B ) ) )  ->  ( A O B )  =  ( A { <. <. 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 ) ) >.
) } B ) )
6 simprl 732 . . . 4  |-  ( ( 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 )  /\  B  e.  ( Morphism SetCat `  U ) ) )
7 domcatval 25930 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  A )  =  ( ( 1st  o.  1st ) `  A )
)
87adantrr 697 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
) )  ->  (
( dom SetCat `  U
) `  A )  =  ( ( 1st 
o.  1st ) `  A
) )
9 codcatval 25936 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  B  e.  ( Morphism SetCat `  U )
)  ->  ( ( cod
SetCat `  U ) `  B )  =  ( ( 2nd  o.  1st ) `  B )
)
109adantrl 696 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
) )  ->  (
( cod SetCat `  U
) `  B )  =  ( ( 2nd 
o.  1st ) `  B
) )
118, 10eqeq12d 2297 . . . . . 6  |-  ( ( 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 )
) )
1211biimpd 198 . . . . 5  |-  ( ( 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
) ) )
1312impr 602 . . . 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 ) `  A
)  =  ( ( 2nd  o.  1st ) `  B ) )
14 3simpa 952 . . . . . 6  |-  ( ( 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 )
) )
15 opex 4237 . . . . . . 7  |-  <. <. (
( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >.  e.  _V
16 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( a  =  A  /\  b  =  B )  ->  b  =  B )
1716fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 1st  o.  1st ) `  b )  =  ( ( 1st 
o.  1st ) `  B
) )
18 simpl 443 . . . . . . . . . . . . . . 15  |-  ( ( a  =  A  /\  b  =  B )  ->  a  =  A )
1918fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 2nd  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  A
) )
2017, 19opeq12d 3804 . . . . . . . . . . . . 13  |-  ( ( a  =  A  /\  b  =  B )  -> 
<. ( ( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >.  =  <. (
( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. )
2118fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( a  =  A  /\  b  =  B )  ->  ( 2nd `  a
)  =  ( 2nd `  A ) )
2216fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( a  =  A  /\  b  =  B )  ->  ( 2nd `  b
)  =  ( 2nd `  B ) )
2321, 22coeq12d 4848 . . . . . . . . . . . . 13  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 2nd `  a
)  o.  ( 2nd `  b ) )  =  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) )
2420, 23opeq12d 3804 . . . . . . . . . . . 12  |-  ( ( a  =  A  /\  b  =  B )  -> 
<. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.  =  <. <. ( ( 1st 
o.  1st ) `  B
) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
)
2524eqeq2d 2294 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  B )  ->  ( c  =  <. <.
( ( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >.  <->  c  =  <. <. ( ( 1st 
o.  1st ) `  B
) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
) )
2625biimp3ar 1282 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  B  /\  c  =  <. <. (
( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )  ->  c  =  <. <. (
( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >. )
2726biantrud 493 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  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
) )  <->  ( (
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 ) ) >.
) ) )
2818eleq1d 2349 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a  e.  (
Morphism
SetCat `  U )  <->  A  e.  ( Morphism SetCat `  U )
) )
2916eleq1d 2349 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  B )  ->  ( b  e.  (
Morphism
SetCat `  U )  <->  B  e.  ( Morphism SetCat `  U )
) )
3018fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 1st  o.  1st ) `  a )  =  ( ( 1st 
o.  1st ) `  A
) )
3116fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 2nd  o.  1st ) `  b )  =  ( ( 2nd 
o.  1st ) `  B
) )
3230, 31eqeq12d 2297 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( ( 1st 
o.  1st ) `  a
)  =  ( ( 2nd  o.  1st ) `  b )  <->  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd  o.  1st ) `  B )
) )
3328, 29, 323anbi123d 1252 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 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
) ) ) )
34333adant3 975 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  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
) )  <->  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) ) ) )
3527, 34bitr3d 246 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  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 ) ) >.
)  <->  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) ) ) )
36 moeq 2941 . . . . . . . . 9  |-  E* c 
c  =  <. <. (
( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >.
3736moani 2195 . . . . . . . 8  |-  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 ) ) >.
)
38 eqid 2283 . . . . . . . 8  |-  { <. <.
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 ) ) >.
) }
3935, 37, 38ovigg 5968 . . . . . . 7  |-  ( ( 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 )  ->  (
( A  e.  (
Morphism
SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) )  ->  ( A { <. <. 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 ) ) >.
) } B )  =  <. <. ( ( 1st 
o.  1st ) `  B
) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
) )
4015, 39mp3an3 1266 . . . . . 6  |-  ( ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  ->  ( ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) )  ->  ( A { <. <. 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 ) ) >.
) } B )  =  <. <. ( ( 1st 
o.  1st ) `  B
) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
) )
4114, 40mpcom 32 . . . . 5  |-  ( ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) )  ->  ( A { <. <. 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 ) ) >.
) } B )  =  <. <. ( ( 1st 
o.  1st ) `  B
) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
)
42413expa 1151 . . . 4  |-  ( ( ( A  e.  (
Morphism
SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  /\  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) )  ->  ( A { <. <. 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 ) ) >.
) } B )  =  <. <. ( ( 1st 
o.  1st ) `  B
) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
)
436, 13, 42syl2anc 642 . . 3  |-  ( ( U  e.  Univ  /\  (
( A  e.  (
Morphism
SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  /\  ( ( dom SetCat `  U
) `  A )  =  ( ( cod SetCat `
 U ) `  B ) ) )  ->  ( A { <. <. 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 ) ) >.
) } B )  =  <. <. ( ( 1st 
o.  1st ) `  B
) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
)
445, 43eqtrd 2315 . 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
) ) >. )
45443impb 1147 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 )  =  <. <.
( ( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
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   ` cfv 5255  (class class class)co 5858   {coprab 5859   1stc1st 6120   2ndc2nd 6121   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   cod
SetCatccodcase 25932   ro SetCatcrocase 25955
This theorem is referenced by:  rocatval  25959  cmp2morpcats  25961
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-domcatset 25920  df-codcatset 25933  df-rocatset 25956
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