Theorem curf1 16865

Description: Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
curfval.g	|- G = ( <. C , D >. curryF F )
curfval.a	|- A = ( Base ` C )
curfval.c	|- ( ph -> C e. Cat )
curfval.d	|- ( ph -> D e. Cat )
curfval.f	|- ( ph -> F e. ( ( C X.c D ) Func E ) )
curfval.b	|- B = ( Base ` D )
curf1.x	|- ( ph -> X e. A )
curf1.k	|- K = ( ( 1st ` G ) ` X )
curf1.j	|- J = ( Hom ` D )
curf1.1	|- .1. = ( Id ` C )

Assertion

Ref	Expression
curf1	|- ( ph -> K = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. )

Distinct variable groups:   y, g, z, .1.   y, A   B, g, y, z   C, g, y, z   D, g, y, z   ph, g, y, z   g, E, y, z   g, J   g, K, y, z   g, X, y, z   g, F, y, z

Allowed substitution hints:   A( z, g)   G( y, z, g)   J( y, z)

Proof of Theorem curf1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curf1.k	. 2 |- K = ( ( 1st ` G ) ` X )
2		curfval.g	. . . 4 |- G = ( <. C , D >. curryF F )
3		curfval.a	. . . 4 |- A = ( Base ` C )
4		curfval.c	. . . 4 |- ( ph -> C e. Cat )
5		curfval.d	. . . 4 |- ( ph -> D e. Cat )
6		curfval.f	. . . 4 |- ( ph -> F e. ( ( C X.c D ) Func E ) )
7		curfval.b	. . . 4 |- B = ( Base ` D )
8		curf1.j	. . . 4 |- J = ( Hom ` D )
9		curf1.1	. . . 4 |- .1. = ( Id ` C )
10	2, 3, 4, 5, 6, 7, 8, 9	curf1fval 16864	. . 3 |- ( ph -> ( 1st ` G ) = ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) )
11		simpr 477	. . . . . 6 |- ( ( ph /\ x = X ) -> x = X )
12	11	oveq1d 6665	. . . . 5 |- ( ( ph /\ x = X ) -> ( x ( 1st ` F ) y ) = ( X ( 1st ` F ) y ) )
13	12	mpteq2dv 4745	. . . 4 |- ( ( ph /\ x = X ) -> ( y e. B |-> ( x ( 1st ` F ) y ) ) = ( y e. B |-> ( X ( 1st ` F ) y ) ) )
14		simp1r 1086	. . . . . . . . 9 |- ( ( ( ph /\ x = X ) /\ y e. B /\ z e. B ) -> x = X )
15	14	opeq1d 4408	. . . . . . . 8 |- ( ( ( ph /\ x = X ) /\ y e. B /\ z e. B ) -> <. x , y >. = <. X , y >. )
16	14	opeq1d 4408	. . . . . . . 8 |- ( ( ( ph /\ x = X ) /\ y e. B /\ z e. B ) -> <. x , z >. = <. X , z >. )
17	15, 16	oveq12d 6668	. . . . . . 7 |- ( ( ( ph /\ x = X ) /\ y e. B /\ z e. B ) -> ( <. x , y >. ( 2nd ` F ) <. x , z >. ) = ( <. X , y >. ( 2nd ` F ) <. X , z >. ) )
18	14	fveq2d 6195	. . . . . . 7 |- ( ( ( ph /\ x = X ) /\ y e. B /\ z e. B ) -> ( .1. ` x ) = ( .1. ` X ) )
19		eqidd 2623	. . . . . . 7 |- ( ( ( ph /\ x = X ) /\ y e. B /\ z e. B ) -> g = g )
20	17, 18, 19	oveq123d 6671	. . . . . 6 |- ( ( ( ph /\ x = X ) /\ y e. B /\ z e. B ) -> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) = ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) )
21	20	mpteq2dv 4745	. . . . 5 |- ( ( ( ph /\ x = X ) /\ y e. B /\ z e. B ) -> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) = ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) )
22	21	mpt2eq3dva 6719	. . . 4 |- ( ( ph /\ x = X ) -> ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) = ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) )
23	13, 22	opeq12d 4410	. . 3 |- ( ( ph /\ x = X ) -> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. )
24		curf1.x	. . 3 |- ( ph -> X e. A )
25		opex 4932	. . . 4 |- <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. e. _V
26	25	a1i 11	. . 3 |- ( ph -> <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. e. _V )
27	10, 23, 24, 26	fvmptd 6288	. 2 |- ( ph -> ( ( 1st ` G ) ` X ) = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. )
28	1, 27	syl5eq 2668	1 |- ( ph -> K = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = 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:  curf11  16866  curf12  16867  curf1cl  16868  curf2ndf  16887