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Theorem hof2fval 15086
Description: The morphism part of the Hom functor, for morphisms  <. f ,  g >. : <. X ,  Y >. --> <. Z ,  W >. (which since the first argument is contravariant means morphisms  f : Z --> X and  g : Y --> W), yields a function (a morphism of  SetCat) mapping  h : X --> Y to  g  o.  h  o.  f : Z --> W. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m  |-  M  =  (HomF
`  C )
hofval.c  |-  ( ph  ->  C  e.  Cat )
hof1.b  |-  B  =  ( Base `  C
)
hof1.h  |-  H  =  ( Hom  `  C
)
hof1.x  |-  ( ph  ->  X  e.  B )
hof1.y  |-  ( ph  ->  Y  e.  B )
hof2.z  |-  ( ph  ->  Z  e.  B )
hof2.w  |-  ( ph  ->  W  e.  B )
hof2.o  |-  .x.  =  (comp `  C )
Assertion
Ref Expression
hof2fval  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  M
) <. Z ,  W >. )  =  ( f  e.  ( Z H X ) ,  g  e.  ( Y H W )  |->  ( h  e.  ( X H Y )  |->  ( ( g ( <. X ,  Y >.  .x.  W )
h ) ( <. Z ,  X >.  .x. 
W ) f ) ) ) )
Distinct variable groups:    f, g, h, B    ph, f, g, h    C, f, g, h   
f, H, g, h   
f, W, g, h    .x. , f, g, h    f, X, g, h    f, Y, g, h    f, Z, g, h
Allowed substitution hints:    M( f, g, h)

Proof of Theorem hof2fval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . . 4  |-  M  =  (HomF
`  C )
2 hofval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
3 hof1.b . . . 4  |-  B  =  ( Base `  C
)
4 hof1.h . . . 4  |-  H  =  ( Hom  `  C
)
5 hof2.o . . . 4  |-  .x.  =  (comp `  C )
61, 2, 3, 4, 5hofval 15083 . . 3  |-  ( ph  ->  M  =  <. ( Hom f  `  C ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y ) )  |->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >. )
7 fvex 5722 . . . 4  |-  ( Hom f  `  C )  e.  _V
8 fvex 5722 . . . . . . 7  |-  ( Base `  C )  e.  _V
93, 8eqeltri 2513 . . . . . 6  |-  B  e. 
_V
109, 9xpex 6529 . . . . 5  |-  ( B  X.  B )  e. 
_V
1110, 10mpt2ex 6671 . . . 4  |-  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( h  e.  ( H `
 x )  |->  ( ( g ( x 
.x.  ( 2nd `  y
) ) h ) ( <. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) )  e.  _V
127, 11op2ndd 6609 . . 3  |-  ( M  =  <. ( Hom f  `  C ) ,  ( x  e.  ( B  X.  B
) ,  y  e.  ( B  X.  B
)  |->  ( f  e.  ( ( 1st `  y
) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >.  ->  ( 2nd `  M )  =  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y ) )  |->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) )
136, 12syl 16 . 2  |-  ( ph  ->  ( 2nd `  M
)  =  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( h  e.  ( H `
 x )  |->  ( ( g ( x 
.x.  ( 2nd `  y
) ) h ) ( <. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) )
14 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  y  =  <. Z ,  W >. )
1514fveq2d 5716 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st `  y )  =  ( 1st `  <. Z ,  W >. ) )
16 hof2.z . . . . . . 7  |-  ( ph  ->  Z  e.  B )
17 hof2.w . . . . . . 7  |-  ( ph  ->  W  e.  B )
18 op1stg 6610 . . . . . . 7  |-  ( ( Z  e.  B  /\  W  e.  B )  ->  ( 1st `  <. Z ,  W >. )  =  Z )
1916, 17, 18syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 1st `  <. Z ,  W >. )  =  Z )
2019adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st ` 
<. Z ,  W >. )  =  Z )
2115, 20eqtrd 2475 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st `  y )  =  Z )
22 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  x  =  <. X ,  Y >. )
2322fveq2d 5716 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st `  x )  =  ( 1st `  <. X ,  Y >. ) )
24 hof1.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
25 hof1.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
26 op1stg 6610 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2724, 25, 26syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
2827adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st ` 
<. X ,  Y >. )  =  X )
2923, 28eqtrd 2475 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st `  x )  =  X )
3021, 29oveq12d 6130 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( ( 1st `  y ) H ( 1st `  x
) )  =  ( Z H X ) )
3122fveq2d 5716 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd `  x )  =  ( 2nd `  <. X ,  Y >. ) )
32 op2ndg 6611 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
3324, 25, 32syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
3433adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd ` 
<. X ,  Y >. )  =  Y )
3531, 34eqtrd 2475 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd `  x )  =  Y )
3614fveq2d 5716 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd `  y )  =  ( 2nd `  <. Z ,  W >. ) )
37 op2ndg 6611 . . . . . . 7  |-  ( ( Z  e.  B  /\  W  e.  B )  ->  ( 2nd `  <. Z ,  W >. )  =  W )
3816, 17, 37syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. Z ,  W >. )  =  W )
3938adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd ` 
<. Z ,  W >. )  =  W )
4036, 39eqtrd 2475 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd `  y )  =  W )
4135, 40oveq12d 6130 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( ( 2nd `  x ) H ( 2nd `  y
) )  =  ( Y H W ) )
4222fveq2d 5716 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( H `  x )  =  ( H `  <. X ,  Y >. ) )
43 df-ov 6115 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
4442, 43syl6eqr 2493 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( H `  x )  =  ( X H Y ) )
4521, 29opeq12d 4088 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  <. ( 1st `  y ) ,  ( 1st `  x )
>.  =  <. Z ,  X >. )
4645, 40oveq12d 6130 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( <. ( 1st `  y ) ,  ( 1st `  x
) >.  .x.  ( 2nd `  y ) )  =  ( <. Z ,  X >.  .x.  W ) )
4722, 40oveq12d 6130 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( x  .x.  ( 2nd `  y
) )  =  (
<. X ,  Y >.  .x. 
W ) )
4847oveqd 6129 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( g
( x  .x.  ( 2nd `  y ) ) h )  =  ( g ( <. X ,  Y >.  .x.  W )
h ) )
49 eqidd 2444 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  f  =  f )
5046, 48, 49oveq123d 6133 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( (
g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f )  =  ( ( g ( <. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) f ) )
5144, 50mpteq12dv 4391 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) )  =  ( h  e.  ( X H Y ) 
|->  ( ( g (
<. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) f ) ) )
5230, 41, 51mpt2eq123dv 6169 . 2  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( f  e.  ( ( 1st `  y
) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) )  =  ( f  e.  ( Z H X ) ,  g  e.  ( Y H W )  |->  ( h  e.  ( X H Y )  |->  ( ( g ( <. X ,  Y >.  .x.  W ) h ) ( <. Z ,  X >.  .x.  W )
f ) ) ) )
53 opelxpi 4892 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
5424, 25, 53syl2anc 661 . 2  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
55 opelxpi 4892 . . 3  |-  ( ( Z  e.  B  /\  W  e.  B )  -> 
<. Z ,  W >.  e.  ( B  X.  B
) )
5616, 17, 55syl2anc 661 . 2  |-  ( ph  -> 
<. Z ,  W >.  e.  ( B  X.  B
) )
57 ovex 6137 . . . 4  |-  ( Z H X )  e. 
_V
58 ovex 6137 . . . 4  |-  ( Y H W )  e. 
_V
5957, 58mpt2ex 6671 . . 3  |-  ( f  e.  ( Z H X ) ,  g  e.  ( Y H W )  |->  ( h  e.  ( X H Y )  |->  ( ( g ( <. X ,  Y >.  .x.  W )
h ) ( <. Z ,  X >.  .x. 
W ) f ) ) )  e.  _V
6059a1i 11 . 2  |-  ( ph  ->  ( f  e.  ( Z H X ) ,  g  e.  ( Y H W ) 
|->  ( h  e.  ( X H Y ) 
|->  ( ( g (
<. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) f ) ) )  e. 
_V )
6113, 52, 54, 56, 60ovmpt2d 6239 1  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  M
) <. Z ,  W >. )  =  ( f  e.  ( Z H X ) ,  g  e.  ( Y H W )  |->  ( h  e.  ( X H Y )  |->  ( ( g ( <. X ,  Y >.  .x.  W )
h ) ( <. Z ,  X >.  .x. 
W ) f ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2993   <.cop 3904    e. cmpt 4371    X. cxp 4859   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   1stc1st 6596   2ndc2nd 6597   Basecbs 14195   Hom chom 14270  compcco 14271   Catccat 14623   Hom f chomf 14625  HomFchof 15079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-hof 15081
This theorem is referenced by:  hof2val  15087
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