Theorem curfpropd 16873

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.

Step	Hyp	Ref	Expression
1		curfpropd.1	. . . . 5 |- ( ph -> ( Hom f ` A ) = ( Hom f ` B ) )
2	1	homfeqbas 16356	. . . 4 |- ( ph -> ( Base ` A ) = ( Base ` B ) )
3		curfpropd.3	. . . . . . . 8 |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
4	3	homfeqbas 16356	. . . . . . 7 |- ( ph -> ( Base ` C ) = ( Base ` D ) )
5	4	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( Base ` A ) ) -> ( Base ` C ) = ( Base ` D ) )
6	5	mpteq1d 4738	. . . . 5 |- ( ( ph /\ x e. ( Base ` A ) ) -> ( y e. ( Base ` C ) |-> ( x ( 1st ` F ) y ) ) = ( y e. ( Base ` D ) |-> ( x ( 1st ` F ) y ) ) )
7	5	adantr 481	. . . . . 6 |- ( ( ( ph /\ x e. ( Base ` A ) ) /\ y e. ( Base ` C ) ) -> ( Base ` C ) = ( Base ` D ) )
8		eqid 2622	. . . . . . . 8 |- ( Base ` C ) = ( Base ` C )
9		eqid 2622	. . . . . . . 8 |- ( Hom ` C ) = ( Hom ` C )
10		eqid 2622	. . . . . . . 8 |- ( Hom ` D ) = ( Hom ` D )
11	3	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. ( Base ` A ) ) /\ ( y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
12		simprl 794	. . . . . . . 8 |- ( ( ( ph /\ x e. ( Base ` A ) ) /\ ( y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
13		simprr 796	. . . . . . . 8 |- ( ( ( ph /\ x e. ( Base ` A ) ) /\ ( y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> z e. ( Base ` C ) )
14	8, 9, 10, 11, 12, 13	homfeqval 16357	. . . . . . 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 )
18	1, 15, 16, 17	cidpropd 16370	. . . . . . . . . 10 |- ( ph -> ( Id ` A ) = ( Id ` B ) )
19	18	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( Base ` A ) ) /\ ( y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> ( Id ` A ) = ( Id ` B ) )
20	19	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ x e. ( Base ` A ) ) /\ ( y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> ( ( Id ` A ) ` x ) = ( ( Id ` B ) ` x ) )
21	20	oveq1d 6665	. . . . . . 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 ) )
22	14, 21	mpteq12dv 4733	. . . . . 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 ) ) )
23	5, 7, 22	mpt2eq123dva 6716	. . . . 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 ) ) ) )
24	6, 23	opeq12d 4410	. . . 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 ) ) ) >. )
25	2, 24	mpteq12dva 4732	. . 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 ) ) ) >. ) )
26	2	adantr 481	. . . 4 |- ( ( ph /\ x e. ( Base ` A ) ) -> ( Base ` A ) = ( Base ` B ) )
27		eqid 2622	. . . . . 6 |- ( Base ` A ) = ( Base ` A )
28		eqid 2622	. . . . . 6 |- ( Hom ` A ) = ( Hom ` A )
29		eqid 2622	. . . . . 6 |- ( Hom ` B ) = ( Hom ` B )
30	1	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` A ) /\ y e. ( Base ` A ) ) ) -> ( Hom f ` A ) = ( Hom f ` B ) )
31		simprl 794	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` A ) /\ y e. ( Base ` A ) ) ) -> x e. ( Base ` A ) )
32		simprr 796	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` A ) /\ y e. ( Base ` A ) ) ) -> y e. ( Base ` A ) )
33	27, 28, 29, 30, 31, 32	homfeqval 16357	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` A ) /\ y e. ( Base ` A ) ) ) -> ( x ( Hom ` A ) y ) = ( x ( Hom ` B ) y ) )
34	4	ad2antrr 762	. . . . . 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 )
38	3, 35, 36, 37	cidpropd 16370	. . . . . . . . 9 |- ( ph -> ( Id ` C ) = ( Id ` D ) )
39	38	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ ( x e. ( Base ` A ) /\ y e. ( Base ` A ) ) ) /\ g e. ( x ( Hom ` A ) y ) ) /\ z e. ( Base ` C ) ) -> ( Id ` C ) = ( Id ` D ) )
40	39	fveq1d 6193	. . . . . . 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 ) )
41	40	oveq2d 6666	. . . . . 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 ) ) )
42	34, 41	mpteq12dva 4732	. . . . 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 ) ) ) )
43	33, 42	mpteq12dva 4732	. . . 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 ) ) ) ) )
44	2, 26, 43	mpt2eq123dva 6716	. . 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 ) ) ) ) ) )
45	25, 44	opeq12d 4410	. 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 2622	. . 3 |- ( <. A , C >. curryF F ) = ( <. A , C >. curryF F )
47		curfpropd.f	. . 3 |- ( ph -> F e. ( ( A X.c C ) Func E ) )
48		eqid 2622	. . 3 |- ( Id ` A ) = ( Id ` A )
49		eqid 2622	. . 3 |- ( Id ` C ) = ( Id ` C )
50	46, 27, 16, 36, 47, 8, 9, 48, 28, 49	curfval 16863	. 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 2622	. . 3 |- ( <. B , D >. curryF F ) = ( <. B , D >. curryF F )
52		eqid 2622	. . 3 |- ( Base ` B ) = ( Base ` B )
53	1, 15, 3, 35, 16, 17, 36, 37	xpcpropd 16848	. . . . 5 |- ( ph -> ( A X.c C ) = ( B X.c D ) )
54	53	oveq1d 6665	. . . 4 |- ( ph -> ( ( A X.c C ) Func E ) = ( ( B X.c D ) Func E ) )
55	47, 54	eleqtrd 2703	. . 3 |- ( ph -> F e. ( ( B X.c D ) Func E ) )
56		eqid 2622	. . 3 |- ( Base ` D ) = ( Base ` D )
57		eqid 2622	. . 3 |- ( Id ` B ) = ( Id ` B )
58		eqid 2622	. . 3 |- ( Id ` D ) = ( Id ` D )
59	51, 52, 17, 37, 55, 56, 10, 57, 29, 58	curfval 16863	. 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 ) ) ) ) ) >. )
60	45, 50, 59	3eqtr4d 2666	1 |- ( ph -> ( <. A , C >. curryF F ) = ( <. B , D >. curryF F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Catccat 16325   Idccid 16326   Hom _f chomf 16327  comp_fccomf 16328   Func cfunc 16514   X._c cxpc 16808   curry_F ccurf 16850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-homf 16331  df-comf 16332  df-xpc 16812  df-curf 16854

This theorem is referenced by:  yonpropd  16908  oppcyon  16909