Theorem curf12 16867

Description: The partially evaluated curry functor at a morphism. (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 )
curf11.y	|- ( ph -> Y e. B )
curf12.j	|- J = ( Hom ` D )
curf12.1	|- .1. = ( Id ` C )
curf12.y	|- ( ph -> Z e. B )
curf12.g	|- ( ph -> H e. ( Y J Z ) )

Assertion

Ref	Expression
curf12	|- ( ph -> ( ( Y ( 2nd ` K ) Z ) ` H ) = ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` F ) <. X , Z >. ) H ) )

Proof of Theorem curf12

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

Step	Hyp	Ref	Expression
1		curfval.g	. . . 4 |- G = ( <. C , D >. curryF F )
2		curfval.a	. . . 4 |- A = ( Base ` C )
3		curfval.c	. . . 4 |- ( ph -> C e. Cat )
4		curfval.d	. . . 4 |- ( ph -> D e. Cat )
5		curfval.f	. . . 4 |- ( ph -> F e. ( ( C X.c D ) Func E ) )
6		curfval.b	. . . 4 |- B = ( Base ` D )
7		curf1.x	. . . 4 |- ( ph -> X e. A )
8		curf1.k	. . . 4 |- K = ( ( 1st ` G ) ` X )
9		curf12.j	. . . 4 |- J = ( Hom ` D )
10		curf12.1	. . . 4 |- .1. = ( Id ` C )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curf1 16865	. . 3 |- ( 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 ) ) ) >. )
12		fvex 6201	. . . . . 6 |- ( Base ` D ) e. _V
13	6, 12	eqeltri 2697	. . . . 5 |- B e. _V
14	13	mptex 6486	. . . 4 |- ( y e. B |-> ( X ( 1st ` F ) y ) ) e. _V
15	13, 13	mpt2ex 7247	. . . 4 |- ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) e. _V
16	14, 15	op2ndd 7179	. . 3 |- ( 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 ) ) ) >. -> ( 2nd ` K ) = ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) )
17	11, 16	syl 17	. 2 |- ( ph -> ( 2nd ` K ) = ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) )
18		curf11.y	. . 3 |- ( ph -> Y e. B )
19		curf12.y	. . . 4 |- ( ph -> Z e. B )
20	19	adantr 481	. . 3 |- ( ( ph /\ y = Y ) -> Z e. B )
21		ovex 6678	. . . . 5 |- ( y J z ) e. _V
22	21	mptex 6486	. . . 4 |- ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) e. _V
23	22	a1i 11	. . 3 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) e. _V )
24		curf12.g	. . . . . 6 |- ( ph -> H e. ( Y J Z ) )
25	24	adantr 481	. . . . 5 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> H e. ( Y J Z ) )
26		simprl 794	. . . . . 6 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> y = Y )
27		simprr 796	. . . . . 6 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> z = Z )
28	26, 27	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> ( y J z ) = ( Y J Z ) )
29	25, 28	eleqtrrd 2704	. . . 4 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> H e. ( y J z ) )
30		ovexd 6680	. . . 4 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) e. _V )
31		simplrl 800	. . . . . . 7 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> y = Y )
32	31	opeq2d 4409	. . . . . 6 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> <. X , y >. = <. X , Y >. )
33		simplrr 801	. . . . . . 7 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> z = Z )
34	33	opeq2d 4409	. . . . . 6 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> <. X , z >. = <. X , Z >. )
35	32, 34	oveq12d 6668	. . . . 5 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> ( <. X , y >. ( 2nd ` F ) <. X , z >. ) = ( <. X , Y >. ( 2nd ` F ) <. X , Z >. ) )
36		eqidd 2623	. . . . 5 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> ( .1. ` X ) = ( .1. ` X ) )
37		simpr 477	. . . . 5 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> g = H )
38	35, 36, 37	oveq123d 6671	. . . 4 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) = ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` F ) <. X , Z >. ) H ) )
39	29, 30, 38	fvmptdv2 6298	. . 3 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> ( ( Y ( 2nd ` K ) Z ) = ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) -> ( ( Y ( 2nd ` K ) Z ) ` H ) = ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` F ) <. X , Z >. ) H ) ) )
40	18, 20, 23, 39	ovmpt2dv 6793	. 2 |- ( ph -> ( ( 2nd ` K ) = ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) -> ( ( Y ( 2nd ` K ) Z ) ` H ) = ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` F ) <. X , Z >. ) H ) ) )
41	17, 40	mpd 15	1 |- ( ph -> ( ( Y ( 2nd ` K ) Z ) ` H ) = ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` F ) <. X , Z >. ) H ) )

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:  curf1cl  16868  curf2cl  16871  uncfcurf  16879  diag12  16884  yon12  16905