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Theorem curfpropd 15720
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 15203 . . . 4  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
3 curfpropd.3 . . . . . . . 8  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
43homfeqbas 15203 . . . . . . 7  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  A )
)  ->  ( Base `  C )  =  (
Base `  D )
)
65mpteq1d 4538 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  A )
)  ->  ( y  e.  ( Base `  C
)  |->  ( x ( 1st `  F ) y ) )  =  ( y  e.  (
Base `  D )  |->  ( x ( 1st `  F ) y ) ) )
75adantr 465 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  C
) )  ->  ( Base `  C )  =  ( Base `  D
) )
8 eqid 2457 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
9 eqid 2457 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
10 eqid 2457 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
113ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
( Hom f  `  C )  =  ( Hom f  `  D ) )
12 simprl 756 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
y  e.  ( Base `  C ) )
13 simprr 757 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
z  e.  ( Base `  C ) )
148, 9, 10, 11, 12, 13homfeqval 15204 . . . . . . 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 15217 . . . . . . . . . 10  |-  ( ph  ->  ( Id `  A
)  =  ( Id
`  B ) )
1918ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
( Id `  A
)  =  ( Id
`  B ) )
2019fveq1d 5874 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Base `  A
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  -> 
( ( Id `  A ) `  x
)  =  ( ( Id `  B ) `
 x ) )
2120oveq1d 6311 . . . . . . 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 4535 . . . . . 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 6357 . . . . 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 4227 . . . 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 4534 . . 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 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  A )
)  ->  ( Base `  A )  =  (
Base `  B )
)
27 eqid 2457 . . . . . 6  |-  ( Base `  A )  =  (
Base `  A )
28 eqid 2457 . . . . . 6  |-  ( Hom  `  A )  =  ( Hom  `  A )
29 eqid 2457 . . . . . 6  |-  ( Hom  `  B )  =  ( Hom  `  B )
301adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  A
)  /\  y  e.  ( Base `  A )
) )  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
31 simprl 756 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  A
)  /\  y  e.  ( Base `  A )
) )  ->  x  e.  ( Base `  A
) )
32 simprr 757 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  A
)  /\  y  e.  ( Base `  A )
) )  ->  y  e.  ( Base `  A
) )
3327, 28, 29, 30, 31, 32homfeqval 15204 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  A
)  /\  y  e.  ( Base `  A )
) )  ->  (
x ( Hom  `  A
) y )  =  ( x ( Hom  `  B ) y ) )
344ad2antrr 725 . . . . . 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 15217 . . . . . . . . 9  |-  ( ph  ->  ( Id `  C
)  =  ( Id
`  D ) )
3938ad3antrrr 729 . . . . . . . 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 5874 . . . . . . 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 6312 . . . . . 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 4534 . . . . 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 4534 . . . 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 6357 . . 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 4227 . 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 2457 . . 3  |-  ( <. A ,  C >. curryF  F )  =  ( <. A ,  C >. curryF  F )
47 curfpropd.f . . 3  |-  ( ph  ->  F  e.  ( ( A  X.c  C )  Func  E
) )
48 eqid 2457 . . 3  |-  ( Id
`  A )  =  ( Id `  A
)
49 eqid 2457 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
5046, 27, 16, 36, 47, 8, 9, 48, 28, 49curfval 15710 . 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 2457 . . 3  |-  ( <. B ,  D >. curryF  F )  =  ( <. B ,  D >. curryF  F )
52 eqid 2457 . . 3  |-  ( Base `  B )  =  (
Base `  B )
531, 15, 3, 35, 16, 17, 36, 37xpcpropd 15695 . . . . 5  |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D
) )
5453oveq1d 6311 . . . 4  |-  ( ph  ->  ( ( A  X.c  C
)  Func  E )  =  ( ( B  X.c  D )  Func  E
) )
5547, 54eleqtrd 2547 . . 3  |-  ( ph  ->  F  e.  ( ( B  X.c  D )  Func  E
) )
56 eqid 2457 . . 3  |-  ( Base `  D )  =  (
Base `  D )
57 eqid 2457 . . 3  |-  ( Id
`  B )  =  ( Id `  B
)
58 eqid 2457 . . 3  |-  ( Id
`  D )  =  ( Id `  D
)
5951, 52, 17, 37, 55, 56, 10, 57, 29, 58curfval 15710 . 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 2508 1  |-  ( ph  ->  ( <. A ,  C >. curryF  F
)  =  ( <. B ,  D >. curryF  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   <.cop 4038    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   Basecbs 14735   Hom chom 14814   Catccat 15172   Idccid 15173   Hom f chomf 15174  compfccomf 15175    Func cfunc 15361    X.c cxpc 15655   curryF ccurf 15697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14737  df-ndx 14738  df-slot 14739  df-base 14740  df-hom 14827  df-cco 14828  df-cat 15176  df-cid 15177  df-homf 15178  df-comf 15179  df-xpc 15659  df-curf 15701
This theorem is referenced by:  yonpropd  15755  oppcyon  15756
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