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Theorem funcres2c 15884
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 392 . . 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 391 . . 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 2471 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
7 eqid 2471 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
8 eqid 2471 . . . . . . 7  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
9 funcres2c.d . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
10 inss2 3644 . . . . . . . 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 15833 . . . . . 6  |-  ( ph  ->  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) )  e.  (Subcat `  D
) )
1312adantr 472 . . . . 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 15676 . . . . . . 7  |-  ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
15 xpss12 4945 . . . . . . . 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 686 . . . . . . 7  |-  ( ( S  i^i  ( Base `  D ) )  X.  ( S  i^i  ( Base `  D ) ) )  C_  ( ( Base `  D )  X.  ( Base `  D
) )
17 fnssres 5699 . . . . . . 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 686 . . . . . 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 472 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F : A --> S )
22 ffn 5739 . . . . . . 7  |-  ( F : A --> S  ->  F  Fn  A )
2321, 22syl 17 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F  Fn  A
)
24 frn 5747 . . . . . . . 8  |-  ( F : A --> S  ->  ran  F  C_  S )
2521, 24syl 17 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  S
)
26 simpr 468 . . . . . . . . . 10  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  F ( C  Func  D ) G )
275, 7, 26funcf1 15849 . . . . . . . . 9  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  F : A
--> ( Base `  D
) )
28 frn 5747 . . . . . . . . 9  |-  ( F : A --> ( Base `  D )  ->  ran  F 
C_  ( Base `  D
) )
2927, 28syl 17 . . . . . . . 8  |-  ( (
ph  /\  F ( C  Func  D ) G )  ->  ran  F  C_  ( Base `  D )
)
30 eqid 2471 . . . . . . . . . . 11  |-  ( Base `  E )  =  (
Base `  E )
31 simpr 468 . . . . . . . . . . 11  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F ( C  Func  E ) G )
325, 30, 31funcf1 15849 . . . . . . . . . 10  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  F : A
--> ( Base `  E
) )
33 frn 5747 . . . . . . . . . 10  |-  ( F : A --> ( Base `  E )  ->  ran  F 
C_  ( Base `  E
) )
3432, 33syl 17 . . . . . . . . 9  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  ran  F  C_  ( Base `  E )
)
35 funcres2c.e . . . . . . . . . 10  |-  E  =  ( Ds  S )
3635, 7ressbasss 15259 . . . . . . . . 9  |-  ( Base `  E )  C_  ( Base `  D )
3734, 36syl6ss 3430 . . . . . . . 8  |-  ( (
ph  /\  F ( C  Func  E ) G )  ->  ran  F  C_  ( Base `  D )
)
3829, 37jaodan 802 . . . . . . 7  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  ( Base `  D ) )
3925, 38ssind 3647 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ran  F  C_  ( S  i^i  ( Base `  D
) ) )
40 df-f 5593 . . . . . 6  |-  ( F : A --> ( S  i^i  ( Base `  D
) )  <->  ( F  Fn  A  /\  ran  F  C_  ( S  i^i  ( Base `  D ) ) ) )
4123, 39, 40sylanbrc 677 . . . . 5  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  F : A --> ( S  i^i  ( Base `  D ) ) )
42 eqid 2471 . . . . . . . . 9  |-  ( Hom  `  D )  =  ( Hom  `  D )
43 simpr 468 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  F ( C  Func  D ) G )
44 simplrl 778 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  x  e.  A )
45 simplrr 779 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  D ) G )  ->  y  e.  A )
465, 6, 42, 43, 44, 45funcf2 15851 . . . . . . . 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 2471 . . . . . . . . . 10  |-  ( Hom  `  E )  =  ( Hom  `  E )
48 simpr 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  F ( C  Func  E ) G )
49 simplrl 778 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  x  e.  A )
50 simplrr 779 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  y  e.  A )
515, 6, 47, 48, 49, 50funcf2 15851 . . . . . . . . 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 15394 . . . . . . . . . . . . 13  |-  ( S  e.  V  ->  ( Hom  `  D )  =  ( Hom  `  E
) )
5452, 53syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( Hom  `  D
)  =  ( Hom  `  E ) )
5554ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A )
)  /\  F ( C  Func  E ) G )  ->  ( Hom  `  D )  =  ( Hom  `  E )
)
5655oveqd 6325 . . . . . . . . . 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 5726 . . . . . . . . 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 240 . . . . . . . 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 802 . . . . . . 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 821 . . . . . 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 472 . . . . . . . . . 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 772 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  ->  x  e.  A )
6361, 62ffvelrnd 6038 . . . . . . . . 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 774 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G ) )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
y  e.  A )
6561, 64ffvelrnd 6038 . . . . . . . . 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 6456 . . . . . . . 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 3416 . . . . . . . . 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 3416 . . . . . . . . 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 15675 . . . . . . . 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 2505 . . . . . . 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 5726 . . . . . 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 240 . . . . 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 15880 . . . 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 2472 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( Hom f  `  C )  =  ( Hom f  `  C ) )
75 eqidd 2472 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  (compf `  C )  =  (compf `  C ) )
767ressinbas 15263 . . . . . . . . . . 11  |-  ( S  e.  V  ->  ( Ds  S )  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
7752, 76syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( Ds  S )  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
7835, 77syl5eq 2517 . . . . . . . . 9  |-  ( ph  ->  E  =  ( Ds  ( S  i^i  ( Base `  D ) ) ) )
7978fveq2d 5883 . . . . . . . 8  |-  ( ph  ->  ( Hom f  `  E )  =  ( Hom f  `  ( Ds  ( S  i^i  ( Base `  D
) ) ) ) )
80 eqid 2471 . . . . . . . . . 10  |-  ( Ds  ( S  i^i  ( Base `  D ) ) )  =  ( Ds  ( S  i^i  ( Base `  D
) ) )
81 eqid 2471 . . . . . . . . . 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 15834 . . . . . . . . 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 466 . . . . . . . 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 2505 . . . . . . 7  |-  ( ph  ->  ( Hom f  `  E )  =  ( Hom f  `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8584adantr 472 . . . . . 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 5883 . . . . . . . 8  |-  ( ph  ->  (compf `  E )  =  (compf `  ( Ds  ( S  i^i  ( Base `  D )
) ) ) )
8782simprd 470 . . . . . . . 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 2505 . . . . . . 7  |-  ( ph  ->  (compf `  E )  =  (compf `  ( D  |`cat  ( ( Hom f  `  D )  |`  ( ( S  i^i  ( Base `  D )
)  X.  ( S  i^i  ( Base `  D
) ) ) ) ) ) )
8988adantr 472 . . . . . 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 4396 . . . . . . . . . . 11  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
91 funcrcl 15846 . . . . . . . . . . 11  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
9290, 91sylbi 200 . . . . . . . . . 10  |-  ( F ( C  Func  D
) G  ->  ( C  e.  Cat  /\  D  e.  Cat ) )
9392simpld 466 . . . . . . . . 9  |-  ( F ( C  Func  D
) G  ->  C  e.  Cat )
94 df-br 4396 . . . . . . . . . . 11  |-  ( F ( C  Func  E
) G  <->  <. F ,  G >.  e.  ( C 
Func  E ) )
95 funcrcl 15846 . . . . . . . . . . 11  |-  ( <. F ,  G >.  e.  ( C  Func  E
)  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
9694, 95sylbi 200 . . . . . . . . . 10  |-  ( F ( C  Func  E
) G  ->  ( C  e.  Cat  /\  E  e.  Cat ) )
9796simpld 466 . . . . . . . . 9  |-  ( F ( C  Func  E
) G  ->  C  e.  Cat )
9893, 97jaoi 386 . . . . . . . 8  |-  ( ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G )  ->  C  e.  Cat )
99 elex 3040 . . . . . . . 8  |-  ( C  e.  Cat  ->  C  e.  _V )
10098, 99syl 17 . . . . . . 7  |-  ( ( F ( C  Func  D ) G  \/  F
( C  Func  E
) G )  ->  C  e.  _V )
101100adantl 473 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  C  e.  _V )
102 ovex 6336 . . . . . . . 8  |-  ( Ds  S )  e.  _V
10335, 102eqeltri 2545 . . . . . . 7  |-  E  e. 
_V
104103a1i 11 . . . . . 6  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  E  e.  _V )
105 ovex 6336 . . . . . . 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 15883 . . . . 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 4406 . . . 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 264 . . 3  |-  ( (
ph  /\  ( F
( C  Func  D
) G  \/  F
( C  Func  E
) G ) )  ->  ( F ( C  Func  D ) G 
<->  F ( C  Func  E ) G ) )
110109ex 441 . 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 361 1  |-  ( ph  ->  ( F ( C 
Func  D ) G  <->  F ( C  Func  E ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389    C_ wss 3390   <.cop 3965   class class class wbr 4395    X. cxp 4837   ran crn 4840    |` cres 4841    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199   ↾s cress 15200   Hom chom 15279   Catccat 15648   Hom f chomf 15650  compfccomf 15651    |`cat cresc 15791  Subcatcsubc 15792    Func cfunc 15837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-hom 15292  df-cco 15293  df-cat 15652  df-cid 15653  df-homf 15654  df-comf 15655  df-ssc 15793  df-resc 15794  df-subc 15795  df-func 15841
This theorem is referenced by:  fthres2c  15914  fullres2c  15922
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