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Theorem funcres2c 14807
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 2441 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
7 eqid 2441 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
8 eqid 2441 . . . . . . 7  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
9 funcres2c.d . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
10 inss2 3568 . . . . . . . 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 14756 . . . . . 6  |-  ( ph  ->  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) )  e.  (Subcat `  D
) )
1312adantr 462 . . . . 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 14628 . . . . . . 7  |-  ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
15 xpss12 4941 . . . . . . . 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 667 . . . . . . 7  |-  ( ( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D ) ) )  C_  ( ( Base `  D )  X.  ( Base `  D
) )
17 fnssres 5521 . . . . . . 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 667 . . . . . 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 462 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F : A --> S )
22 ffn 5556 . . . . . . 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 5562 . . . . . . . 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 458 . . . . . . . . . 10  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  F ( C  Func  D ) G )
275, 7, 26funcf1 14772 . . . . . . . . 9  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  F : A
--> ( Base `  D
) )
28 frn 5562 . . . . . . . . 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 2441 . . . . . . . . . . 11  |-  ( Base `  E )  =  (
Base `  E )
31 simpr 458 . . . . . . . . . . 11  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F ( C  Func  E ) G )
325, 30, 31funcf1 14772 . . . . . . . . . 10  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F : A
--> ( Base `  E
) )
33 frn 5562 . . . . . . . . . 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 14226 . . . . . . . . 9  |-  ( Base `  E )  C_  ( Base `  D )
3734, 36syl6ss 3365 . . . . . . . 8  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  ran  F  C_  ( Base `  D )
)
3829, 37jaodan 778 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  ( Base `  D ) )
3925, 38ssind 3571 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  ( S  i^i  ( Base `  D
) ) )
40 df-f 5419 . . . . . 6  |-  ( F : A --> ( S  i^i  ( Base `  D
) )  <->  ( F  Fn  A  /\  ran  F  C_  ( S  i^i  ( Base `  D ) ) ) )
4123, 39, 40sylanbrc 659 . . . . 5  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F : A --> ( S  i^i  ( Base `  D ) ) )
42 eqid 2441 . . . . . . . . 9  |-  ( Hom  `  D )  =  ( Hom  `  D )
43 simpr 458 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  F ( C  Func  D ) G )
44 simplrl 754 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  x  e.  A )
45 simplrr 755 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  y  e.  A )
465, 6, 42, 43, 44, 45funcf2 14774 . . . . . . . 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 2441 . . . . . . . . . 10  |-  ( Hom  `  E )  =  ( Hom  `  E )
48 simpr 458 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  F ( C  Func  E ) G )
49 simplrl 754 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  x  e.  A )
50 simplrr 755 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  y  e.  A )
515, 6, 47, 48, 49, 50funcf2 14774 . . . . . . . . 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 2442 . . . . . . . . . 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 14353 . . . . . . . . . . . . 13  |-  ( S  e.  V  ->  ( Hom  `  D )  =  ( Hom  `  E
) )
5553, 54syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( Hom  `  D
)  =  ( Hom  `  E ) )
5655ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  ( Hom  `  D )  =  ( Hom  `  E )
)
5756oveqd 6107 . . . . . . . . . 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 5551 . . . . . . . . 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 778 . . . . . . 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 797 . . . . . 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 2442 . . . . . . 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 462 . . . . . . . . . 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 750 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  ->  x  e.  A )
6563, 64ffvelrnd 5841 . . . . . . . . 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 751 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
y  e.  A )
6763, 66ffvelrnd 5841 . . . . . . . . 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 6230 . . . . . . . 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 3351 . . . . . . . . 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 3351 . . . . . . . . 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 14627 . . . . . . . 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 2473 . . . . . . 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 5551 . . . . . 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 14803 . . . 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 2442 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( Hom f  `  C )  =  ( Hom f  `  C ) )
77 eqidd 2442 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  (compf `  C )  =  (compf `  C ) )
787ressinbas 14230 . . . . . . . . . . 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 2485 . . . . . . . . 9  |-  ( ph  ->  E  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
8180fveq2d 5692 . . . . . . . 8  |-  ( ph  ->  ( Hom f  `  E )  =  ( Hom f  `  ( Ds  ( S  i^i  ( Base `  D
) ) ) ) )
82 eqid 2441 . . . . . . . . . 10  |-  ( Ds  ( S  i^i  ( Base `  D ) ) )  =  ( Ds  ( S  i^i  ( Base `  D
) ) )
83 eqid 2441 . . . . . . . . . 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 14757 . . . . . . . . 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 456 . . . . . . . 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 2473 . . . . . . 7  |-  ( ph  ->  ( Hom f  `  E )  =  ( Hom f  `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8786adantr 462 . . . . . 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 5692 . . . . . . . 8  |-  ( ph  ->  (compf `  E )  =  (compf `  ( Ds  ( S  i^i  ( Base `  D )
) ) ) )
8984simprd 460 . . . . . . . 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 2473 . . . . . . 7  |-  ( ph  ->  (compf `  E )  =  (compf `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
9190adantr 462 . . . . . 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 4290 . . . . . . . . . . 11  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
93 funcrcl 14769 . . . . . . . . . . 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 456 . . . . . . . . 9  |-  ( F ( C  Func  D
) G  ->  C  e.  Cat )
96 df-br 4290 . . . . . . . . . . 11  |-  ( F ( C  Func  E
) G  <->  <. F ,  G >.  e.  ( C 
Func  E ) )
97 funcrcl 14769 . . . . . . . . . . 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 456 . . . . . . . . 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 2979 . . . . . . . 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 463 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  C  e.  _V )
104 ovex 6115 . . . . . . . 8  |-  ( Ds  S )  e.  _V
10535, 104eqeltri 2511 . . . . . . 7  |-  E  e. 
_V
106105a1i 11 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  E  e.  _V )
107 ovex 6115 . . . . . . 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 14806 . . . . 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 4300 . . . 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 1364    e. wcel 1761   _Vcvv 2970    i^i cin 3324    C_ wss 3325   <.cop 3880   class class class wbr 4289    X. cxp 4834   ran crn 4837    |` cres 4838    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   ↾s cress 14171   Hom chom 14245   Catccat 14598   Hom f chomf 14600  compfccomf 14601    |`cat cresc 14717  Subcatcsubc 14718    Func cfunc 14760
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-ssc 14719  df-resc 14720  df-subc 14721  df-func 14764
This theorem is referenced by:  fthres2c  14837  fullres2c  14845
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