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Theorem curfpropd 16118
Description: If two categories have the same set of objects, morphisms, and compositions, then they curry the same functor to the same result. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
curfpropd.1  |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
curfpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
curfpropd.3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
curfpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
curfpropd.a  |-  ( ph  ->  A  e.  Cat )
curfpropd.b  |-  ( ph  ->  B  e.  Cat )
curfpropd.c  |-  ( ph  ->  C  e.  Cat )
curfpropd.d  |-  ( ph  ->  D  e.  Cat )
curfpropd.f  |-  ( ph  ->  F  e.  ( ( A  X.c  C )  Func  E
) )
Assertion
Ref Expression
curfpropd  |-  ( ph  ->  ( <. A ,  C >. curryF  F
)  =  ( <. B ,  D >. curryF  F ) )

Proof of Theorem curfpropd
Dummy variables  x  g  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfpropd.1 . . . . 5  |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
21homfeqbas 15601 . . . 4  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
3 curfpropd.3 . . . . . . . 8  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
43homfeqbas 15601 . . . . . . 7  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
54adantr 467 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  A )
)  ->  ( Base `  C )  =  (
Base `  D )
)
65mpteq1d 4484 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  A )
)  ->  ( y  e.  ( Base `  C
)  |->  ( x ( 1st `  F ) y ) )  =  ( y  e.  (
Base `  D )  |->  ( x ( 1st `  F ) y ) ) )
75adantr 467 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  C
) )  ->  ( Base `  C )  =  ( Base `  D
) )
8 eqid 2451 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
9 eqid 2451 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
10 eqid 2451 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
113ad2antrr 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
( Hom f  `  C )  =  ( Hom f  `  D ) )
12 simprl 764 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
y  e.  ( Base `  C ) )
13 simprr 766 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
z  e.  ( Base `  C ) )
148, 9, 10, 11, 12, 13homfeqval 15602 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
( y ( Hom  `  C ) z )  =  ( y ( Hom  `  D )
z ) )
15 curfpropd.2 . . . . . . . . . . 11  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
16 curfpropd.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  Cat )
17 curfpropd.b . . . . . . . . . . 11  |-  ( ph  ->  B  e.  Cat )
181, 15, 16, 17cidpropd 15615 . . . . . . . . . 10  |-  ( ph  ->  ( Id `  A
)  =  ( Id
`  B ) )
1918ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
( Id `  A
)  =  ( Id
`  B ) )
2019fveq1d 5867 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
( ( Id `  A ) `  x
)  =  ( ( Id `  B ) `
 x ) )
2120oveq1d 6305 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
( ( ( Id
`  A ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g )  =  ( ( ( Id
`  B ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) )
2214, 21mpteq12dv 4481 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
( g  e.  ( y ( Hom  `  C
) z )  |->  ( ( ( Id `  A ) `  x
) ( <. x ,  y >. ( 2nd `  F ) <.
x ,  z >.
) g ) )  =  ( g  e.  ( y ( Hom  `  D ) z ) 
|->  ( ( ( Id
`  B ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) )
235, 7, 22mpt2eq123dva 6352 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  A )
)  ->  ( y  e.  ( Base `  C
) ,  z  e.  ( Base `  C
)  |->  ( g  e.  ( y ( Hom  `  C ) z ) 
|->  ( ( ( Id
`  A ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) )  =  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  ( g  e.  ( y ( Hom  `  D ) z ) 
|->  ( ( ( Id
`  B ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) )
246, 23opeq12d 4174 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  A )
)  ->  <. ( y  e.  ( Base `  C
)  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  (
Base `  C ) ,  z  e.  ( Base `  C )  |->  ( g  e.  ( y ( Hom  `  C
) z )  |->  ( ( ( Id `  A ) `  x
) ( <. x ,  y >. ( 2nd `  F ) <.
x ,  z >.
) g ) ) ) >.  =  <. ( y  e.  ( Base `  D )  |->  ( x ( 1st `  F
) y ) ) ,  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  ( g  e.  ( y ( Hom  `  D ) z ) 
|->  ( ( ( Id
`  B ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. )
252, 24mpteq12dva 4480 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  A )  |-> 
<. ( y  e.  (
Base `  C )  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  ( Base `  C
) ,  z  e.  ( Base `  C
)  |->  ( g  e.  ( y ( Hom  `  C ) z ) 
|->  ( ( ( Id
`  A ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. )  =  ( x  e.  ( Base `  B )  |->  <. (
y  e.  ( Base `  D )  |->  ( x ( 1st `  F
) y ) ) ,  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  ( g  e.  ( y ( Hom  `  D ) z ) 
|->  ( ( ( Id
`  B ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) )
262adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  A )
)  ->  ( Base `  A )  =  (
Base `  B )
)
27 eqid 2451 . . . . . 6  |-  ( Base `  A )  =  (
Base `  A )
28 eqid 2451 . . . . . 6  |-  ( Hom  `  A )  =  ( Hom  `  A )
29 eqid 2451 . . . . . 6  |-  ( Hom  `  B )  =  ( Hom  `  B )
301adantr 467 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  A
)  /\  y  e.  ( Base `  A )
) )  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
31 simprl 764 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  A
)  /\  y  e.  ( Base `  A )
) )  ->  x  e.  ( Base `  A
) )
32 simprr 766 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  A
)  /\  y  e.  ( Base `  A )
) )  ->  y  e.  ( Base `  A
) )
3327, 28, 29, 30, 31, 32homfeqval 15602 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  A
)  /\  y  e.  ( Base `  A )
) )  ->  (
x ( Hom  `  A
) y )  =  ( x ( Hom  `  B ) y ) )
344ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  A )  /\  y  e.  ( Base `  A
) ) )  /\  g  e.  ( x
( Hom  `  A ) y ) )  -> 
( Base `  C )  =  ( Base `  D
) )
35 curfpropd.4 . . . . . . . . . 10  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
36 curfpropd.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  Cat )
37 curfpropd.d . . . . . . . . . 10  |-  ( ph  ->  D  e.  Cat )
383, 35, 36, 37cidpropd 15615 . . . . . . . . 9  |-  ( ph  ->  ( Id `  C
)  =  ( Id
`  D ) )
3938ad3antrrr 736 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  A )  /\  y  e.  ( Base `  A ) ) )  /\  g  e.  ( x ( Hom  `  A
) y ) )  /\  z  e.  (
Base `  C )
)  ->  ( Id `  C )  =  ( Id `  D ) )
4039fveq1d 5867 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  A )  /\  y  e.  ( Base `  A ) ) )  /\  g  e.  ( x ( Hom  `  A
) y ) )  /\  z  e.  (
Base `  C )
)  ->  ( ( Id `  C ) `  z )  =  ( ( Id `  D
) `  z )
)
4140oveq2d 6306 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  A )  /\  y  e.  ( Base `  A ) ) )  /\  g  e.  ( x ( Hom  `  A
) y ) )  /\  z  e.  (
Base `  C )
)  ->  ( g
( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  C ) `  z ) )  =  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  D ) `  z ) ) )
4234, 41mpteq12dva 4480 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( Base `  A )  /\  y  e.  ( Base `  A
) ) )  /\  g  e.  ( x
( Hom  `  A ) y ) )  -> 
( z  e.  (
Base `  C )  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  C ) `  z ) ) )  =  ( z  e.  ( Base `  D
)  |->  ( g (
<. x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  D ) `  z ) ) ) )
4333, 42mpteq12dva 4480 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  A
)  /\  y  e.  ( Base `  A )
) )  ->  (
g  e.  ( x ( Hom  `  A
) y )  |->  ( z  e.  ( Base `  C )  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  C ) `  z ) ) ) )  =  ( g  e.  ( x ( Hom  `  B )
y )  |->  ( z  e.  ( Base `  D
)  |->  ( g (
<. x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  D ) `  z ) ) ) ) )
442, 26, 43mpt2eq123dva 6352 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  A ) ,  y  e.  ( Base `  A )  |->  ( g  e.  ( x ( Hom  `  A
) y )  |->  ( z  e.  ( Base `  C )  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  C ) `  z ) ) ) ) )  =  ( x  e.  ( Base `  B ) ,  y  e.  ( Base `  B
)  |->  ( g  e.  ( x ( Hom  `  B ) y ) 
|->  ( z  e.  (
Base `  D )  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  D ) `  z ) ) ) ) ) )
4525, 44opeq12d 4174 . 2  |-  ( ph  -> 
<. ( x  e.  (
Base `  A )  |-> 
<. ( y  e.  (
Base `  C )  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  ( Base `  C
) ,  z  e.  ( Base `  C
)  |->  ( g  e.  ( y ( Hom  `  C ) z ) 
|->  ( ( ( Id
`  A ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  ( Base `  A ) ,  y  e.  ( Base `  A
)  |->  ( g  e.  ( x ( Hom  `  A ) y ) 
|->  ( z  e.  (
Base `  C )  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  C ) `  z ) ) ) ) ) >.  =  <. ( x  e.  ( Base `  B )  |->  <. (
y  e.  ( Base `  D )  |->  ( x ( 1st `  F
) y ) ) ,  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  ( g  e.  ( y ( Hom  `  D ) z ) 
|->  ( ( ( Id
`  B ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  ( Base `  B ) ,  y  e.  ( Base `  B
)  |->  ( g  e.  ( x ( Hom  `  B ) y ) 
|->  ( z  e.  (
Base `  D )  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  D ) `  z ) ) ) ) ) >. )
46 eqid 2451 . . 3  |-  ( <. A ,  C >. curryF  F )  =  ( <. A ,  C >. curryF  F )
47 curfpropd.f . . 3  |-  ( ph  ->  F  e.  ( ( A  X.c  C )  Func  E
) )
48 eqid 2451 . . 3  |-  ( Id
`  A )  =  ( Id `  A
)
49 eqid 2451 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
5046, 27, 16, 36, 47, 8, 9, 48, 28, 49curfval 16108 . 2  |-  ( ph  ->  ( <. A ,  C >. curryF  F
)  =  <. (
x  e.  ( Base `  A )  |->  <. (
y  e.  ( Base `  C )  |->  ( x ( 1st `  F
) y ) ) ,  ( y  e.  ( Base `  C
) ,  z  e.  ( Base `  C
)  |->  ( g  e.  ( y ( Hom  `  C ) z ) 
|->  ( ( ( Id
`  A ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  ( Base `  A ) ,  y  e.  ( Base `  A
)  |->  ( g  e.  ( x ( Hom  `  A ) y ) 
|->  ( z  e.  (
Base `  C )  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  C ) `  z ) ) ) ) ) >. )
51 eqid 2451 . . 3  |-  ( <. B ,  D >. curryF  F )  =  ( <. B ,  D >. curryF  F )
52 eqid 2451 . . 3  |-  ( Base `  B )  =  (
Base `  B )
531, 15, 3, 35, 16, 17, 36, 37xpcpropd 16093 . . . . 5  |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D
) )
5453oveq1d 6305 . . . 4  |-  ( ph  ->  ( ( A  X.c  C
)  Func  E )  =  ( ( B  X.c  D )  Func  E
) )
5547, 54eleqtrd 2531 . . 3  |-  ( ph  ->  F  e.  ( ( B  X.c  D )  Func  E
) )
56 eqid 2451 . . 3  |-  ( Base `  D )  =  (
Base `  D )
57 eqid 2451 . . 3  |-  ( Id
`  B )  =  ( Id `  B
)
58 eqid 2451 . . 3  |-  ( Id
`  D )  =  ( Id `  D
)
5951, 52, 17, 37, 55, 56, 10, 57, 29, 58curfval 16108 . 2  |-  ( ph  ->  ( <. B ,  D >. curryF  F
)  =  <. (
x  e.  ( Base `  B )  |->  <. (
y  e.  ( Base `  D )  |->  ( x ( 1st `  F
) y ) ) ,  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  ( g  e.  ( y ( Hom  `  D ) z ) 
|->  ( ( ( Id
`  B ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  ( Base `  B ) ,  y  e.  ( Base `  B
)  |->  ( g  e.  ( x ( Hom  `  B ) y ) 
|->  ( z  e.  (
Base `  D )  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( ( Id
`  D ) `  z ) ) ) ) ) >. )
6045, 50, 593eqtr4d 2495 1  |-  ( ph  ->  ( <. A ,  C >. curryF  F
)  =  ( <. B ,  D >. curryF  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   <.cop 3974    |-> cmpt 4461   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   1stc1st 6791   2ndc2nd 6792   Basecbs 15121   Hom chom 15201   Catccat 15570   Idccid 15571   Hom f chomf 15572  compfccomf 15573    Func cfunc 15759    X.c cxpc 16053   curryF ccurf 16095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-hom 15214  df-cco 15215  df-cat 15574  df-cid 15575  df-homf 15576  df-comf 15577  df-xpc 16057  df-curf 16099
This theorem is referenced by:  yonpropd  16153  oppcyon  16154
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