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Theorem cofth 15162
Description: The composition of two faithful functors is faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
cofth.f  |-  ( ph  ->  F  e.  ( C Faith 
D ) )
cofth.g  |-  ( ph  ->  G  e.  ( D Faith 
E ) )
Assertion
Ref Expression
cofth  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Faith  E ) )

Proof of Theorem cofth
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 15089 . . 3  |-  Rel  ( C  Func  E )
2 fthfunc 15134 . . . . 5  |-  ( C Faith 
D )  C_  ( C  Func  D )
3 cofth.f . . . . 5  |-  ( ph  ->  F  e.  ( C Faith 
D ) )
42, 3sseldi 3502 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
5 fthfunc 15134 . . . . 5  |-  ( D Faith 
E )  C_  ( D  Func  E )
6 cofth.g . . . . 5  |-  ( ph  ->  G  e.  ( D Faith 
E ) )
75, 6sseldi 3502 . . . 4  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
84, 7cofucl 15115 . . 3  |-  ( ph  ->  ( G  o.func  F )  e.  ( C  Func  E
) )
9 1st2nd 6830 . . 3  |-  ( ( Rel  ( C  Func  E )  /\  ( G  o.func 
F )  e.  ( C  Func  E )
)  ->  ( G  o.func  F )  =  <. ( 1st `  ( G  o.func  F ) ) ,  ( 2nd `  ( G  o.func  F )
) >. )
101, 8, 9sylancr 663 . 2  |-  ( ph  ->  ( G  o.func  F )  =  <. ( 1st `  ( G  o.func 
F ) ) ,  ( 2nd `  ( G  o.func 
F ) ) >.
)
11 1st2ndbr 6833 . . . . 5  |-  ( ( Rel  ( C  Func  E )  /\  ( G  o.func 
F )  e.  ( C  Func  E )
)  ->  ( 1st `  ( G  o.func  F )
) ( C  Func  E ) ( 2nd `  ( G  o.func 
F ) ) )
121, 8, 11sylancr 663 . . . 4  |-  ( ph  ->  ( 1st `  ( G  o.func 
F ) ) ( C  Func  E )
( 2nd `  ( G  o.func 
F ) ) )
13 eqid 2467 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
14 eqid 2467 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
15 eqid 2467 . . . . . . . 8  |-  ( Hom  `  E )  =  ( Hom  `  E )
16 relfth 15136 . . . . . . . . 9  |-  Rel  ( D Faith  E )
176adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  G  e.  ( D Faith  E ) )
18 1st2ndbr 6833 . . . . . . . . 9  |-  ( ( Rel  ( D Faith  E
)  /\  G  e.  ( D Faith  E ) )  ->  ( 1st `  G
) ( D Faith  E
) ( 2nd `  G
) )
1916, 17, 18sylancr 663 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  G ) ( D Faith  E ) ( 2nd `  G ) )
20 eqid 2467 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
21 relfunc 15089 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
224adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F  e.  ( C  Func  D
) )
23 1st2ndbr 6833 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2421, 22, 23sylancr 663 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2520, 13, 24funcf1 15093 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) : ( Base `  C
) --> ( Base `  D
) )
26 simprl 755 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
2725, 26ffvelrnd 6022 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
28 simprr 756 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
2925, 28ffvelrnd 6022 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
3013, 14, 15, 19, 27, 29fthf1 15144 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) ) : ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) -1-1-> ( ( ( 1st `  G ) `
 ( ( 1st `  F ) `  x
) ) ( Hom  `  E ) ( ( 1st `  G ) `
 ( ( 1st `  F ) `  y
) ) ) )
31 eqid 2467 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
32 relfth 15136 . . . . . . . . 9  |-  Rel  ( C Faith  D )
333adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F  e.  ( C Faith  D ) )
34 1st2ndbr 6833 . . . . . . . . 9  |-  ( ( Rel  ( C Faith  D
)  /\  F  e.  ( C Faith  D ) )  ->  ( 1st `  F
) ( C Faith  D
) ( 2nd `  F
) )
3532, 33, 34sylancr 663 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C Faith  D ) ( 2nd `  F ) )
3620, 31, 14, 35, 26, 28fthf1 15144 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  F
) y ) : ( x ( Hom  `  C ) y )
-1-1-> ( ( ( 1st `  F ) `  x
) ( Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
37 f1co 5790 . . . . . . 7  |-  ( ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) ) : ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) -1-1-> ( ( ( 1st `  G ) `
 ( ( 1st `  F ) `  x
) ) ( Hom  `  E ) ( ( 1st `  G ) `
 ( ( 1st `  F ) `  y
) ) )  /\  ( x ( 2nd `  F ) y ) : ( x ( Hom  `  C )
y ) -1-1-> ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) )  ->  (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) : ( x ( Hom  `  C )
y ) -1-1-> ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) ( Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) ) )
3830, 36, 37syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) : ( x ( Hom  `  C )
y ) -1-1-> ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) ( Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) ) )
397adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  G  e.  ( D  Func  E
) )
4020, 22, 39, 26, 28cofu2nd 15112 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  ( G  o.func 
F ) ) y )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )
41 eqidd 2468 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( Hom  `  C
) y )  =  ( x ( Hom  `  C ) y ) )
4220, 22, 39, 26cofu1 15111 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  ( G  o.func 
F ) ) `  x )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) )
4320, 22, 39, 28cofu1 15111 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  ( G  o.func 
F ) ) `  y )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) )
4442, 43oveq12d 6302 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  ( G  o.func 
F ) ) `  x ) ( Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  =  ( ( ( 1st `  G ) `  (
( 1st `  F
) `  x )
) ( Hom  `  E
) ( ( 1st `  G ) `  (
( 1st `  F
) `  y )
) ) )
4540, 41, 44f1eq123d 5811 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( x ( 2nd `  ( G  o.func  F )
) y ) : ( x ( Hom  `  C ) y )
-1-1-> ( ( ( 1st `  ( G  o.func  F )
) `  x )
( Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  <->  ( (
( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) : ( x ( Hom  `  C )
y ) -1-1-> ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) ( Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) ) ) )
4638, 45mpbird 232 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  ( G  o.func 
F ) ) y ) : ( x ( Hom  `  C
) y ) -1-1-> ( ( ( 1st `  ( G  o.func 
F ) ) `  x ) ( Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) ) )
4746ralrimivva 2885 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ( x ( 2nd `  ( G  o.func 
F ) ) y ) : ( x ( Hom  `  C
) y ) -1-1-> ( ( ( 1st `  ( G  o.func 
F ) ) `  x ) ( Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) ) )
4820, 31, 15isfth2 15142 . . . 4  |-  ( ( 1st `  ( G  o.func 
F ) ) ( C Faith  E ) ( 2nd `  ( G  o.func 
F ) )  <->  ( ( 1st `  ( G  o.func  F ) ) ( C  Func  E ) ( 2nd `  ( G  o.func 
F ) )  /\  A. x  e.  ( Base `  C ) A. y  e.  ( Base `  C
) ( x ( 2nd `  ( G  o.func 
F ) ) y ) : ( x ( Hom  `  C
) y ) -1-1-> ( ( ( 1st `  ( G  o.func 
F ) ) `  x ) ( Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) ) ) )
4912, 47, 48sylanbrc 664 . . 3  |-  ( ph  ->  ( 1st `  ( G  o.func 
F ) ) ( C Faith  E ) ( 2nd `  ( G  o.func 
F ) ) )
50 df-br 4448 . . 3  |-  ( ( 1st `  ( G  o.func 
F ) ) ( C Faith  E ) ( 2nd `  ( G  o.func 
F ) )  <->  <. ( 1st `  ( G  o.func  F )
) ,  ( 2nd `  ( G  o.func  F )
) >.  e.  ( C Faith 
E ) )
5149, 50sylib 196 . 2  |-  ( ph  -> 
<. ( 1st `  ( G  o.func 
F ) ) ,  ( 2nd `  ( G  o.func 
F ) ) >.  e.  ( C Faith  E ) )
5210, 51eqeltrd 2555 1  |-  ( ph  ->  ( G  o.func  F )  e.  ( C Faith  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   <.cop 4033   class class class wbr 4447    o. ccom 5003   Rel wrel 5004   -1-1->wf1 5585   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   Basecbs 14490   Hom chom 14566    Func cfunc 15081    o.func ccofu 15083   Faith cfth 15130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422  df-ixp 7470  df-cat 14923  df-cid 14924  df-func 15085  df-cofu 15087  df-fth 15132
This theorem is referenced by:  coffth  15163
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