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Theorem funcres2c 15119
Description: Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
Hypotheses
Ref Expression
funcres2c.a  |-  A  =  ( Base `  C
)
funcres2c.e  |-  E  =  ( Ds  S )
funcres2c.d  |-  ( ph  ->  D  e.  Cat )
funcres2c.r  |-  ( ph  ->  S  e.  V )
funcres2c.1  |-  ( ph  ->  F : A --> S )
Assertion
Ref Expression
funcres2c  |-  ( ph  ->  ( F ( C 
Func  D ) G  <->  F ( C  Func  E ) G ) )

Proof of Theorem funcres2c
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 385 . . 3  |-  ( F ( C  Func  D
) G  ->  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )
21a1i 11 . 2  |-  ( ph  ->  ( F ( C 
Func  D ) G  -> 
( F ( C 
Func  D ) G  \/  F ( C  Func  E ) G ) ) )
3 olc 384 . . 3  |-  ( F ( C  Func  E
) G  ->  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )
43a1i 11 . 2  |-  ( ph  ->  ( F ( C 
Func  E ) G  -> 
( F ( C 
Func  D ) G  \/  F ( C  Func  E ) G ) ) )
5 funcres2c.a . . . . 5  |-  A  =  ( Base `  C
)
6 eqid 2462 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
7 eqid 2462 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
8 eqid 2462 . . . . . . 7  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
9 funcres2c.d . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
10 inss2 3714 . . . . . . . 8  |-  ( S  i^i  ( Base `  D
) )  C_  ( Base `  D )
1110a1i 11 . . . . . . 7  |-  ( ph  ->  ( S  i^i  ( Base `  D ) ) 
C_  ( Base `  D
) )
127, 8, 9, 11fullsubc 15068 . . . . . 6  |-  ( ph  ->  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) )  e.  (Subcat `  D
) )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( ( Hom f  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) )  e.  (Subcat `  D )
)
148, 7homffn 14940 . . . . . . 7  |-  ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
15 xpss12 5101 . . . . . . . 8  |-  ( ( ( S  i^i  ( Base `  D ) ) 
C_  ( Base `  D
)  /\  ( S  i^i  ( Base `  D
) )  C_  ( Base `  D ) )  ->  ( ( S  i^i  ( Base `  D
) )  X.  ( S  i^i  ( Base `  D
) ) )  C_  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1610, 10, 15mp2an 672 . . . . . . 7  |-  ( ( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D ) ) )  C_  ( ( Base `  D )  X.  ( Base `  D
) )
17 fnssres 5687 . . . . . . 7  |-  ( ( ( Hom f  `  D )  Fn  ( ( Base `  D
)  X.  ( Base `  D ) )  /\  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) )  C_  ( ( Base `  D
)  X.  ( Base `  D ) ) )  ->  ( ( Hom f  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) )  Fn  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) )
1814, 16, 17mp2an 672 . . . . . 6  |-  ( ( Hom f  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) )  Fn  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) )
1918a1i 11 . . . . 5  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( ( Hom f  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) )  Fn  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) )
20 funcres2c.1 . . . . . . . 8  |-  ( ph  ->  F : A --> S )
2120adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F : A --> S )
22 ffn 5724 . . . . . . 7  |-  ( F : A --> S  ->  F  Fn  A )
2321, 22syl 16 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F  Fn  A
)
24 frn 5730 . . . . . . . 8  |-  ( F : A --> S  ->  ran  F  C_  S )
2521, 24syl 16 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  S
)
26 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  F ( C  Func  D ) G )
275, 7, 26funcf1 15084 . . . . . . . . 9  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  F : A
--> ( Base `  D
) )
28 frn 5730 . . . . . . . . 9  |-  ( F : A --> ( Base `  D )  ->  ran  F 
C_  ( Base `  D
) )
2927, 28syl 16 . . . . . . . 8  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  ran  F  C_  ( Base `  D )
)
30 eqid 2462 . . . . . . . . . . 11  |-  ( Base `  E )  =  (
Base `  E )
31 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F ( C  Func  E ) G )
325, 30, 31funcf1 15084 . . . . . . . . . 10  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F : A
--> ( Base `  E
) )
33 frn 5730 . . . . . . . . . 10  |-  ( F : A --> ( Base `  E )  ->  ran  F 
C_  ( Base `  E
) )
3432, 33syl 16 . . . . . . . . 9  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  ran  F  C_  ( Base `  E )
)
35 funcres2c.e . . . . . . . . . 10  |-  E  =  ( Ds  S )
3635, 7ressbasss 14538 . . . . . . . . 9  |-  ( Base `  E )  C_  ( Base `  D )
3734, 36syl6ss 3511 . . . . . . . 8  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  ran  F  C_  ( Base `  D )
)
3829, 37jaodan 783 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  ( Base `  D ) )
3925, 38ssind 3717 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  ( S  i^i  ( Base `  D
) ) )
40 df-f 5585 . . . . . 6  |-  ( F : A --> ( S  i^i  ( Base `  D
) )  <->  ( F  Fn  A  /\  ran  F  C_  ( S  i^i  ( Base `  D ) ) ) )
4123, 39, 40sylanbrc 664 . . . . 5  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F : A --> ( S  i^i  ( Base `  D ) ) )
42 eqid 2462 . . . . . . . . 9  |-  ( Hom  `  D )  =  ( Hom  `  D )
43 simpr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  F ( C  Func  D ) G )
44 simplrl 759 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  x  e.  A )
45 simplrr 760 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  y  e.  A )
465, 6, 42, 43, 44, 45funcf2 15086 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  ( x G y ) : ( x ( Hom  `  C ) y ) --> ( ( F `  x ) ( Hom  `  D ) ( F `
 y ) ) )
47 eqid 2462 . . . . . . . . . 10  |-  ( Hom  `  E )  =  ( Hom  `  E )
48 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  F ( C  Func  E ) G )
49 simplrl 759 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  x  e.  A )
50 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  y  e.  A )
515, 6, 47, 48, 49, 50funcf2 15086 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  ( x G y ) : ( x ( Hom  `  C ) y ) --> ( ( F `  x ) ( Hom  `  E ) ( F `
 y ) ) )
52 eqidd 2463 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  ( x
( Hom  `  C ) y )  =  ( x ( Hom  `  C
) y ) )
53 funcres2c.r . . . . . . . . . . . . 13  |-  ( ph  ->  S  e.  V )
5435, 42resshom 14665 . . . . . . . . . . . . 13  |-  ( S  e.  V  ->  ( Hom  `  D )  =  ( Hom  `  E
) )
5553, 54syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( Hom  `  D
)  =  ( Hom  `  E ) )
5655ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  ( Hom  `  D )  =  ( Hom  `  E )
)
5756oveqd 6294 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  ( ( F `  x )
( Hom  `  D ) ( F `  y
) )  =  ( ( F `  x
) ( Hom  `  E
) ( F `  y ) ) )
5852, 57feq23d 5719 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  ( (
x G y ) : ( x ( Hom  `  C )
y ) --> ( ( F `  x ) ( Hom  `  D
) ( F `  y ) )  <->  ( x G y ) : ( x ( Hom  `  C ) y ) --> ( ( F `  x ) ( Hom  `  E ) ( F `
 y ) ) ) )
5951, 58mpbird 232 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  ( x G y ) : ( x ( Hom  `  C ) y ) --> ( ( F `  x ) ( Hom  `  D ) ( F `
 y ) ) )
6046, 59jaodan 783 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( x G y ) : ( x ( Hom  `  C
) y ) --> ( ( F `  x
) ( Hom  `  D
) ( F `  y ) ) )
6160an32s 802 . . . . . 6  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x G y ) : ( x ( Hom  `  C
) y ) --> ( ( F `  x
) ( Hom  `  D
) ( F `  y ) ) )
62 eqidd 2463 . . . . . . 7  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x ( Hom  `  C ) y )  =  ( x ( Hom  `  C )
y ) )
6341adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  ->  F : A --> ( S  i^i  ( Base `  D
) ) )
64 simprl 755 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  ->  x  e.  A )
6563, 64ffvelrnd 6015 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( F `  x
)  e.  ( S  i^i  ( Base `  D
) ) )
66 simprr 756 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
y  e.  A )
6763, 66ffvelrnd 6015 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( F `  y
)  e.  ( S  i^i  ( Base `  D
) ) )
6865, 67ovresd 6420 . . . . . . . 8  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x ) ( ( Hom f  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) ) ( F `  y ) )  =  ( ( F `  x ) ( Hom f  `  D ) ( F `  y ) ) )
6910, 65sseldi 3497 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( F `  x
)  e.  ( Base `  D ) )
7010, 67sseldi 3497 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( F `  y
)  e.  ( Base `  D ) )
718, 7, 42, 69, 70homfval 14939 . . . . . . . 8  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x ) ( Hom f  `  D ) ( F `
 y ) )  =  ( ( F `
 x ) ( Hom  `  D )
( F `  y
) ) )
7268, 71eqtrd 2503 . . . . . . 7  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x ) ( ( Hom f  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) ) ( F `  y ) )  =  ( ( F `  x ) ( Hom  `  D
) ( F `  y ) ) )
7362, 72feq23d 5719 . . . . . 6  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( x G y ) : ( x ( Hom  `  C
) y ) --> ( ( F `  x
) ( ( Hom f  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) ) ( F `  y ) )  <->  ( x G y ) : ( x ( Hom  `  C
) y ) --> ( ( F `  x
) ( Hom  `  D
) ( F `  y ) ) ) )
7461, 73mpbird 232 . . . . 5  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x G y ) : ( x ( Hom  `  C
) y ) --> ( ( F `  x
) ( ( Hom f  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) ) ( F `  y ) ) )
755, 6, 13, 19, 41, 74funcres2b 15115 . . . 4  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( F ( C  Func  D ) G 
<->  F ( C  Func  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) G ) )
76 eqidd 2463 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( Hom f  `  C )  =  ( Hom f  `  C ) )
77 eqidd 2463 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  (compf `  C )  =  (compf `  C ) )
787ressinbas 14542 . . . . . . . . . . 11  |-  ( S  e.  V  ->  ( Ds  S )  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
7953, 78syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( Ds  S )  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
8035, 79syl5eq 2515 . . . . . . . . 9  |-  ( ph  ->  E  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
8180fveq2d 5863 . . . . . . . 8  |-  ( ph  ->  ( Hom f  `  E )  =  ( Hom f  `  ( Ds  ( S  i^i  ( Base `  D
) ) ) ) )
82 eqid 2462 . . . . . . . . . 10  |-  ( Ds  ( S  i^i  ( Base `  D ) ) )  =  ( Ds  ( S  i^i  ( Base `  D
) ) )
83 eqid 2462 . . . . . . . . . 10  |-  ( D  |`cat 
( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) )  =  ( D  |`cat 
( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) )
847, 8, 9, 11, 82, 83fullresc 15069 . . . . . . . . 9  |-  ( ph  ->  ( ( Hom f  `  ( Ds  ( S  i^i  ( Base `  D ) ) ) )  =  ( Hom f  `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) )  /\  (compf `  ( Ds  ( S  i^i  ( Base `  D ) ) ) )  =  (compf `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) ) )
8584simpld 459 . . . . . . . 8  |-  ( ph  ->  ( Hom f  `  ( Ds  ( S  i^i  ( Base `  D
) ) ) )  =  ( Hom f  `  ( D  |`cat 
( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8681, 85eqtrd 2503 . . . . . . 7  |-  ( ph  ->  ( Hom f  `  E )  =  ( Hom f  `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8786adantr 465 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( Hom f  `  E )  =  ( Hom f  `  ( D  |`cat 
( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8880fveq2d 5863 . . . . . . . 8  |-  ( ph  ->  (compf `  E )  =  (compf `  ( Ds  ( S  i^i  ( Base `  D )
) ) ) )
8984simprd 463 . . . . . . . 8  |-  ( ph  ->  (compf `  ( Ds  ( S  i^i  ( Base `  D )
) ) )  =  (compf `  ( D  |`cat  ( ( Hom f  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) ) ) ) )
9088, 89eqtrd 2503 . . . . . . 7  |-  ( ph  ->  (compf `  E )  =  (compf `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
9190adantr 465 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  (compf `  E )  =  (compf `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
92 df-br 4443 . . . . . . . . . . 11  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
93 funcrcl 15081 . . . . . . . . . . 11  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
9492, 93sylbi 195 . . . . . . . . . 10  |-  ( F ( C  Func  D
) G  ->  ( C  e.  Cat  /\  D  e.  Cat ) )
9594simpld 459 . . . . . . . . 9  |-  ( F ( C  Func  D
) G  ->  C  e.  Cat )
96 df-br 4443 . . . . . . . . . . 11  |-  ( F ( C  Func  E
) G  <->  <. F ,  G >.  e.  ( C 
Func  E ) )
97 funcrcl 15081 . . . . . . . . . . 11  |-  ( <. F ,  G >.  e.  ( C  Func  E
)  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
9896, 97sylbi 195 . . . . . . . . . 10  |-  ( F ( C  Func  E
) G  ->  ( C  e.  Cat  /\  E  e.  Cat ) )
9998simpld 459 . . . . . . . . 9  |-  ( F ( C  Func  E
) G  ->  C  e.  Cat )
10095, 99jaoi 379 . . . . . . . 8  |-  ( ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G )  ->  C  e.  Cat )
101 elex 3117 . . . . . . . 8  |-  ( C  e.  Cat  ->  C  e.  _V )
102100, 101syl 16 . . . . . . 7  |-  ( ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G )  ->  C  e.  _V )
103102adantl 466 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  C  e.  _V )
104 ovex 6302 . . . . . . . 8  |-  ( Ds  S )  e.  _V
10535, 104eqeltri 2546 . . . . . . 7  |-  E  e. 
_V
106105a1i 11 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  E  e.  _V )
107 ovex 6302 . . . . . . 7  |-  ( D  |`cat 
( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) )  e.  _V
108107a1i 11 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) )  e.  _V )
10976, 77, 87, 91, 103, 103, 106, 108funcpropd 15118 . . . . 5  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( C  Func  E )  =  ( C 
Func  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
110109breqd 4453 . . . 4  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( F ( C  Func  E ) G 
<->  F ( C  Func  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) G ) )
11175, 110bitr4d 256 . . 3  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( F ( C  Func  D ) G 
<->  F ( C  Func  E ) G ) )
112111ex 434 . 2  |-  ( ph  ->  ( ( F ( C  Func  D ) G  \/  F ( C  Func  E ) G )  ->  ( F
( C  Func  D
) G  <->  F ( C  Func  E ) G ) ) )
1132, 4, 112pm5.21ndd 354 1  |-  ( ph  ->  ( F ( C 
Func  D ) G  <->  F ( C  Func  E ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108    i^i cin 3470    C_ wss 3471   <.cop 4028   class class class wbr 4442    X. cxp 4992   ran crn 4995    |` cres 4996    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   Basecbs 14481   ↾s cress 14482   Hom chom 14557   Catccat 14910   Hom f chomf 14912  compfccomf 14913    |`cat cresc 15029  Subcatcsubc 15030    Func cfunc 15072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-hom 14570  df-cco 14571  df-cat 14914  df-cid 14915  df-homf 14916  df-comf 14917  df-ssc 15031  df-resc 15032  df-subc 15033  df-func 15076
This theorem is referenced by:  fthres2c  15149  fullres2c  15157
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