Theorem curf2 16869

Description: Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
curf2.g	|- G = ( <. C , D >. curryF F )
curf2.a	|- A = ( Base ` C )
curf2.c	|- ( ph -> C e. Cat )
curf2.d	|- ( ph -> D e. Cat )
curf2.f	|- ( ph -> F e. ( ( C X.c D ) Func E ) )
curf2.b	|- B = ( Base ` D )
curf2.h	|- H = ( Hom ` C )
curf2.i	|- I = ( Id ` D )
curf2.x	|- ( ph -> X e. A )
curf2.y	|- ( ph -> Y e. A )
curf2.k	|- ( ph -> K e. ( X H Y ) )
curf2.l	|- L = ( ( X ( 2nd ` G ) Y ) ` K )

Assertion

Ref	Expression
curf2	|- ( ph -> L = ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) )

Distinct variable groups:   z, C   z, F   z, H   z, L   z, E   z, G   z, I   ph, z   z, B   z, D   z, X   z, K   z, Y

Allowed substitution hint:   A( z)

Proof of Theorem curf2

Dummy variables x y g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curf2.l	. 2 |- L = ( ( X ( 2nd ` G ) Y ) ` K )
2		curf2.g	. . . . 5 |- G = ( <. C , D >. curryF F )
3		curf2.a	. . . . 5 |- A = ( Base ` C )
4		curf2.c	. . . . 5 |- ( ph -> C e. Cat )
5		curf2.d	. . . . 5 |- ( ph -> D e. Cat )
6		curf2.f	. . . . 5 |- ( ph -> F e. ( ( C X.c D ) Func E ) )
7		curf2.b	. . . . 5 |- B = ( Base ` D )
8		eqid 2622	. . . . 5 |- ( Hom ` D ) = ( Hom ` D )
9		eqid 2622	. . . . 5 |- ( Id ` C ) = ( Id ` C )
10		curf2.h	. . . . 5 |- H = ( Hom ` C )
11		curf2.i	. . . . 5 |- I = ( Id ` D )
12	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	curfval 16863	. . . 4 |- ( ph -> G = <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. )
13		fvex 6201	. . . . . . 7 |- ( Base ` C ) e. _V
14	3, 13	eqeltri 2697	. . . . . 6 |- A e. _V
15	14	mptex 6486	. . . . 5 |- ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) e. _V
16	14, 14	mpt2ex 7247	. . . . 5 |- ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) e. _V
17	15, 16	op2ndd 7179	. . . 4 |- ( G = <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. -> ( 2nd ` G ) = ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) )
18	12, 17	syl 17	. . 3 |- ( ph -> ( 2nd ` G ) = ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) )
19		curf2.x	. . . 4 |- ( ph -> X e. A )
20		curf2.y	. . . . 5 |- ( ph -> Y e. A )
21	20	adantr 481	. . . 4 |- ( ( ph /\ x = X ) -> Y e. A )
22		ovex 6678	. . . . . 6 |- ( x H y ) e. _V
23	22	mptex 6486	. . . . 5 |- ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) e. _V
24	23	a1i 11	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) e. _V )
25		curf2.k	. . . . . . 7 |- ( ph -> K e. ( X H Y ) )
26	25	adantr 481	. . . . . 6 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> K e. ( X H Y ) )
27		simprl 794	. . . . . . 7 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> x = X )
28		simprr 796	. . . . . . 7 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> y = Y )
29	27, 28	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x H y ) = ( X H Y ) )
30	26, 29	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> K e. ( x H y ) )
31		fvex 6201	. . . . . . . 8 |- ( Base ` D ) e. _V
32	7, 31	eqeltri 2697	. . . . . . 7 |- B e. _V
33	32	mptex 6486	. . . . . 6 |- ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) e. _V
34	33	a1i 11	. . . . 5 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) e. _V )
35		simplrl 800	. . . . . . . . 9 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> x = X )
36	35	opeq1d 4408	. . . . . . . 8 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> <. x , z >. = <. X , z >. )
37		simplrr 801	. . . . . . . . 9 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> y = Y )
38	37	opeq1d 4408	. . . . . . . 8 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> <. y , z >. = <. Y , z >. )
39	36, 38	oveq12d 6668	. . . . . . 7 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> ( <. x , z >. ( 2nd ` F ) <. y , z >. ) = ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) )
40		simpr 477	. . . . . . 7 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> g = K )
41		eqidd 2623	. . . . . . 7 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> ( I ` z ) = ( I ` z ) )
42	39, 40, 41	oveq123d 6671	. . . . . 6 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) = ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) )
43	42	mpteq2dv 4745	. . . . 5 |- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ g = K ) -> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) = ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) )
44	30, 34, 43	fvmptdv2 6298	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( X ( 2nd ` G ) Y ) = ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) -> ( ( X ( 2nd ` G ) Y ) ` K ) = ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) ) )
45	19, 21, 24, 44	ovmpt2dv 6793	. . 3 |- ( ph -> ( ( 2nd ` G ) = ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) -> ( ( X ( 2nd ` G ) Y ) ` K ) = ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) ) )
46	18, 45	mpd 15	. 2 |- ( ph -> ( ( X ( 2nd ` G ) Y ) ` K ) = ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) )
47	1, 46	syl5eq 2668	1 |- ( ph -> L = ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.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   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-curf 16854

This theorem is referenced by:  curf2val  16870  curf2cl  16871  curfcl  16872  diag2  16885  curf2ndf  16887