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Theorem funcres2c 14098
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 375 . . 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 374 . . 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 2436 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
7 eqid 2436 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
8 eqid 2436 . . . . . . 7  |-  (  Homf  `  D )  =  (  Homf 
`  D )
9 funcres2c.d . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
10 inss2 3562 . . . . . . . 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 14047 . . . . . 6  |-  ( ph  ->  ( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) )  e.  (Subcat `  D
) )
1312adantr 452 . . . . 5  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( (  Homf  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) )  e.  (Subcat `  D )
)
148, 7homffn 13919 . . . . . . 7  |-  (  Homf  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
15 xpss12 4981 . . . . . . . 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 654 . . . . . . 7  |-  ( ( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D ) ) )  C_  ( ( Base `  D )  X.  ( Base `  D
) )
17 fnssres 5558 . . . . . . 7  |-  ( ( (  Homf 
`  D )  Fn  ( ( Base `  D
)  X.  ( Base `  D ) )  /\  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) )  C_  ( ( Base `  D
)  X.  ( Base `  D ) ) )  ->  ( (  Homf  `  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 654 . . . . . 6  |-  ( (  Homf 
`  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 ) )  ->  ( (  Homf  `  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 452 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F : A --> S )
22 ffn 5591 . . . . . . 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 5597 . . . . . . . 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 448 . . . . . . . . . 10  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  F ( C  Func  D ) G )
275, 7, 26funcf1 14063 . . . . . . . . 9  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  F : A
--> ( Base `  D
) )
28 frn 5597 . . . . . . . . 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 2436 . . . . . . . . . . 11  |-  ( Base `  E )  =  (
Base `  E )
31 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F ( C  Func  E ) G )
325, 30, 31funcf1 14063 . . . . . . . . . 10  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F : A
--> ( Base `  E
) )
33 frn 5597 . . . . . . . . . 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 13521 . . . . . . . . 9  |-  ( Base `  E )  C_  ( Base `  D )
3734, 36syl6ss 3360 . . . . . . . 8  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  ran  F  C_  ( Base `  D )
)
3829, 37jaodan 761 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  ( Base `  D ) )
3925, 38ssind 3565 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  ( S  i^i  ( Base `  D
) ) )
40 df-f 5458 . . . . . 6  |-  ( F : A --> ( S  i^i  ( Base `  D
) )  <->  ( F  Fn  A  /\  ran  F  C_  ( S  i^i  ( Base `  D ) ) ) )
4123, 39, 40sylanbrc 646 . . . . 5  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F : A --> ( S  i^i  ( Base `  D ) ) )
42 eqid 2436 . . . . . . . . 9  |-  (  Hom  `  D )  =  (  Hom  `  D )
43 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  F ( C  Func  D ) G )
44 simplrl 737 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  x  e.  A )
45 simplrr 738 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  y  e.  A )
465, 6, 42, 43, 44, 45funcf2 14065 . . . . . . . 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 2436 . . . . . . . . . 10  |-  (  Hom  `  E )  =  (  Hom  `  E )
48 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  F ( C  Func  E ) G )
49 simplrl 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  x  e.  A )
50 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  y  e.  A )
515, 6, 47, 48, 49, 50funcf2 14065 . . . . . . . . 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 2437 . . . . . . . . . 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 13646 . . . . . . . . . . . . 13  |-  ( S  e.  V  ->  (  Hom  `  D )  =  (  Hom  `  E
) )
5553, 54syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  (  Hom  `  D
)  =  (  Hom  `  E ) )
5655ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  (  Hom  `  D )  =  (  Hom  `  E )
)
5756oveqd 6098 . . . . . . . . . 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 5588 . . . . . . . . 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 224 . . . . . . . 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 761 . . . . . . 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 780 . . . . . 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 2437 . . . . . . 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 452 . . . . . . . . . 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 733 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  ->  x  e.  A )
6563, 64ffvelrnd 5871 . . . . . . . . 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 734 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
y  e.  A )
6763, 66ffvelrnd 5871 . . . . . . . . 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 6214 . . . . . . . 8  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x ) ( (  Homf 
`  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ( F `  y
) )  =  ( ( F `  x
) (  Homf  `  D ) ( F `  y
) ) )
6910, 65sseldi 3346 . . . . . . . . 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 3346 . . . . . . . . 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 13918 . . . . . . . 8  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x ) (  Homf  `  D ) ( F `
 y ) )  =  ( ( F `
 x ) (  Hom  `  D )
( F `  y
) ) )
7268, 71eqtrd 2468 . . . . . . 7  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x ) ( (  Homf 
`  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ( F `  y
) )  =  ( ( F `  x
) (  Hom  `  D
) ( F `  y ) ) )
7362, 72feq23d 5588 . . . . . 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
) ( (  Homf  `  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 224 . . . . 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
) ( (  Homf  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) ) ( F `  y ) ) )
755, 6, 13, 19, 41, 74funcres2b 14094 . . . 4  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( F ( C  Func  D ) G 
<->  F ( C  Func  ( D  |`cat  ( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) G ) )
76 eqidd 2437 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  (  Homf  `  C )  =  (  Homf  `  C ) )
77 eqidd 2437 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  (compf `  C )  =  (compf `  C ) )
787ressinbas 13525 . . . . . . . . . . 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 2480 . . . . . . . . 9  |-  ( ph  ->  E  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
8180fveq2d 5732 . . . . . . . 8  |-  ( ph  ->  (  Homf 
`  E )  =  (  Homf 
`  ( Ds  ( S  i^i  ( Base `  D
) ) ) ) )
82 eqid 2436 . . . . . . . . . 10  |-  ( Ds  ( S  i^i  ( Base `  D ) ) )  =  ( Ds  ( S  i^i  ( Base `  D
) ) )
83 eqid 2436 . . . . . . . . . 10  |-  ( D  |`cat 
( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) )  =  ( D  |`cat 
( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) )
847, 8, 9, 11, 82, 83fullresc 14048 . . . . . . . . 9  |-  ( ph  ->  ( (  Homf  `  ( Ds  ( S  i^i  ( Base `  D ) ) ) )  =  (  Homf  `  ( D  |`cat  ( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) )  /\  (compf `  ( Ds  ( S  i^i  ( Base `  D ) ) ) )  =  (compf `  ( D  |`cat  ( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) ) )
8584simpld 446 . . . . . . . 8  |-  ( ph  ->  (  Homf 
`  ( Ds  ( S  i^i  ( Base `  D
) ) ) )  =  (  Homf  `  ( D  |`cat 
( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8681, 85eqtrd 2468 . . . . . . 7  |-  ( ph  ->  (  Homf 
`  E )  =  (  Homf 
`  ( D  |`cat  ( (  Homf 
`  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8786adantr 452 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  (  Homf  `  E )  =  (  Homf  `  ( D  |`cat 
( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8880fveq2d 5732 . . . . . . . 8  |-  ( ph  ->  (compf `  E )  =  (compf `  ( Ds  ( S  i^i  ( Base `  D )
) ) ) )
8984simprd 450 . . . . . . . 8  |-  ( ph  ->  (compf `  ( Ds  ( S  i^i  ( Base `  D )
) ) )  =  (compf `  ( D  |`cat  ( (  Homf  `  D )  |`  (
( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D )
) ) ) ) ) )
9088, 89eqtrd 2468 . . . . . . 7  |-  ( ph  ->  (compf `  E )  =  (compf `  ( D  |`cat  ( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
9190adantr 452 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  (compf `  E )  =  (compf `  ( D  |`cat  ( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
92 df-br 4213 . . . . . . . . . . 11  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
93 funcrcl 14060 . . . . . . . . . . 11  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
9492, 93sylbi 188 . . . . . . . . . 10  |-  ( F ( C  Func  D
) G  ->  ( C  e.  Cat  /\  D  e.  Cat ) )
9594simpld 446 . . . . . . . . 9  |-  ( F ( C  Func  D
) G  ->  C  e.  Cat )
96 df-br 4213 . . . . . . . . . . 11  |-  ( F ( C  Func  E
) G  <->  <. F ,  G >.  e.  ( C 
Func  E ) )
97 funcrcl 14060 . . . . . . . . . . 11  |-  ( <. F ,  G >.  e.  ( C  Func  E
)  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
9896, 97sylbi 188 . . . . . . . . . 10  |-  ( F ( C  Func  E
) G  ->  ( C  e.  Cat  /\  E  e.  Cat ) )
9998simpld 446 . . . . . . . . 9  |-  ( F ( C  Func  E
) G  ->  C  e.  Cat )
10095, 99jaoi 369 . . . . . . . 8  |-  ( ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G )  ->  C  e.  Cat )
101 elex 2964 . . . . . . . 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 453 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  C  e.  _V )
104 ovex 6106 . . . . . . . 8  |-  ( Ds  S )  e.  _V
10535, 104eqeltri 2506 . . . . . . 7  |-  E  e. 
_V
106105a1i 11 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  E  e.  _V )
107 ovex 6106 . . . . . . 7  |-  ( D  |`cat 
( (  Homf  `  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  ( (  Homf 
`  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) )  e.  _V )
10976, 77, 87, 91, 103, 103, 106, 108funcpropd 14097 . . . . 5  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( C  Func  E )  =  ( C 
Func  ( D  |`cat  ( (  Homf 
`  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
110109breqd 4223 . . . 4  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( F ( C  Func  E ) G 
<->  F ( C  Func  ( D  |`cat  ( (  Homf  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) G ) )
11175, 110bitr4d 248 . . 3  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( F ( C  Func  D ) G 
<->  F ( C  Func  E ) G ) )
112111ex 424 . 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 344 1  |-  ( ph  ->  ( F ( C 
Func  D ) G  <->  F ( C  Func  E ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   <.cop 3817   class class class wbr 4212    X. cxp 4876   ran crn 4879    |` cres 4880    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470    Hom chom 13540   Catccat 13889    Homf chomf 13891  compfccomf 13892    |`cat cresc 14008  Subcatcsubc 14009    Func cfunc 14051
This theorem is referenced by:  fthres2c  14128  fullres2c  14136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-hom 13553  df-cco 13554  df-cat 13893  df-cid 13894  df-homf 13895  df-comf 13896  df-ssc 14010  df-resc 14011  df-subc 14012  df-func 14055
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