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Theorem funcres2c 15139
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 3701 . . . . . . . 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 15088 . . . . . 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 14960 . . . . . . 7  |-  ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
15 xpss12 5094 . . . . . . . 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 5680 . . . . . . 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 5717 . . . . . . 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 5723 . . . . . . . 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 15104 . . . . . . . . 9  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  F : A
--> ( Base `  D
) )
28 frn 5723 . . . . . . . . 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 461 . . . . . . . . . . 11  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F ( C  Func  E ) G )
325, 30, 31funcf1 15104 . . . . . . . . . 10  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F : A
--> ( Base `  E
) )
33 frn 5723 . . . . . . . . . 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 14561 . . . . . . . . 9  |-  ( Base `  E )  C_  ( Base `  D )
3734, 36syl6ss 3498 . . . . . . . 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 3704 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  ( S  i^i  ( Base `  D
) ) )
40 df-f 5578 . . . . . 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 2441 . . . . . . . . 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 15106 . . . . . . . 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 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 15106 . . . . . . . . 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 funcres2c.r . . . . . . . . . . . . 13  |-  ( ph  ->  S  e.  V )
5335, 42resshom 14688 . . . . . . . . . . . . 13  |-  ( S  e.  V  ->  ( Hom  `  D )  =  ( Hom  `  E
) )
5452, 53syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( Hom  `  D
)  =  ( Hom  `  E ) )
5554ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  ( Hom  `  D )  =  ( Hom  `  E )
)
5655oveqd 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 ) ) )
5756feq3d 5705 . . . . . . . . 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 ) ) ) )
5851, 57mpbird 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 ) ) )
5946, 58jaodan 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 ) ) )
6059an32s 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 ) ) )
6141adantr 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
) ) )
62 simprl 755 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  ->  x  e.  A )
6361, 62ffvelrnd 6013 . . . . . . . . 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
) ) )
64 simprr 756 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
y  e.  A )
6561, 64ffvelrnd 6013 . . . . . . . . 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
) ) )
6663, 65ovresd 6424 . . . . . . . 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 ) ) )
6710, 63sseldi 3484 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( F `  x
)  e.  ( Base `  D ) )
6810, 65sseldi 3484 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( F `  y
)  e.  ( Base `  D ) )
698, 7, 42, 67, 68homfval 14959 . . . . . . . 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
) ) )
7066, 69eqtrd 2482 . . . . . . 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 ) ) )
7170feq3d 5705 . . . . . 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 ) ) ) )
7260, 71mpbird 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 ) ) )
735, 6, 13, 19, 41, 72funcres2b 15135 . . . 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 ) )
74 eqidd 2442 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( Hom f  `  C )  =  ( Hom f  `  C ) )
75 eqidd 2442 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  (compf `  C )  =  (compf `  C ) )
767ressinbas 14565 . . . . . . . . . . 11  |-  ( S  e.  V  ->  ( Ds  S )  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
7752, 76syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( Ds  S )  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
7835, 77syl5eq 2494 . . . . . . . . 9  |-  ( ph  ->  E  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
7978fveq2d 5856 . . . . . . . 8  |-  ( ph  ->  ( Hom f  `  E )  =  ( Hom f  `  ( Ds  ( S  i^i  ( Base `  D
) ) ) ) )
80 eqid 2441 . . . . . . . . . 10  |-  ( Ds  ( S  i^i  ( Base `  D ) ) )  =  ( Ds  ( S  i^i  ( Base `  D
) ) )
81 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
) ) ) ) )
827, 8, 9, 11, 80, 81fullresc 15089 . . . . . . . . 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
) ) ) ) ) ) ) )
8382simpld 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
) ) ) ) ) ) )
8479, 83eqtrd 2482 . . . . . . 7  |-  ( ph  ->  ( Hom f  `  E )  =  ( Hom f  `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8584adantr 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
) ) ) ) ) ) )
8678fveq2d 5856 . . . . . . . 8  |-  ( ph  ->  (compf `  E )  =  (compf `  ( Ds  ( S  i^i  ( Base `  D )
) ) ) )
8782simprd 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 )
) ) ) ) ) )
8886, 87eqtrd 2482 . . . . . . 7  |-  ( ph  ->  (compf `  E )  =  (compf `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8988adantr 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
) ) ) ) ) ) )
90 df-br 4434 . . . . . . . . . . 11  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
91 funcrcl 15101 . . . . . . . . . . 11  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
9290, 91sylbi 195 . . . . . . . . . 10  |-  ( F ( C  Func  D
) G  ->  ( C  e.  Cat  /\  D  e.  Cat ) )
9392simpld 459 . . . . . . . . 9  |-  ( F ( C  Func  D
) G  ->  C  e.  Cat )
94 df-br 4434 . . . . . . . . . . 11  |-  ( F ( C  Func  E
) G  <->  <. F ,  G >.  e.  ( C 
Func  E ) )
95 funcrcl 15101 . . . . . . . . . . 11  |-  ( <. F ,  G >.  e.  ( C  Func  E
)  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
9694, 95sylbi 195 . . . . . . . . . 10  |-  ( F ( C  Func  E
) G  ->  ( C  e.  Cat  /\  E  e.  Cat ) )
9796simpld 459 . . . . . . . . 9  |-  ( F ( C  Func  E
) G  ->  C  e.  Cat )
9893, 97jaoi 379 . . . . . . . 8  |-  ( ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G )  ->  C  e.  Cat )
99 elex 3102 . . . . . . . 8  |-  ( C  e.  Cat  ->  C  e.  _V )
10098, 99syl 16 . . . . . . 7  |-  ( ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G )  ->  C  e.  _V )
101100adantl 466 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  C  e.  _V )
102 ovex 6305 . . . . . . . 8  |-  ( Ds  S )  e.  _V
10335, 102eqeltri 2525 . . . . . . 7  |-  E  e. 
_V
104103a1i 11 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  E  e.  _V )
105 ovex 6305 . . . . . . 7  |-  ( D  |`cat 
( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) )  e.  _V
106105a1i 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 )
10774, 75, 85, 89, 101, 101, 104, 106funcpropd 15138 . . . . 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
) ) ) ) ) ) )
108107breqd 4444 . . . 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 ) )
10973, 108bitr4d 256 . . 3  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( F ( C  Func  D ) G 
<->  F ( C  Func  E ) G ) )
110109ex 434 . 2  |-  ( ph  ->  ( ( F ( C  Func  D ) G  \/  F ( C  Func  E ) G )  ->  ( F
( C  Func  D
) G  <->  F ( C  Func  E ) G ) ) )
1112, 4, 110pm5.21ndd 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 1381    e. wcel 1802   _Vcvv 3093    i^i cin 3457    C_ wss 3458   <.cop 4016   class class class wbr 4433    X. cxp 4983   ran crn 4986    |` cres 4987    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾s cress 14505   Hom chom 14580   Catccat 14933   Hom f chomf 14935  compfccomf 14936    |`cat cresc 15049  Subcatcsubc 15050    Func cfunc 15092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-hom 14593  df-cco 14594  df-cat 14937  df-cid 14938  df-homf 14939  df-comf 14940  df-ssc 15051  df-resc 15052  df-subc 15053  df-func 15096
This theorem is referenced by:  fthres2c  15169  fullres2c  15177
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