Theorem curf12 14329

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 14327	. . 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 5745	. . . . . 6 |- ( Base ` D ) e. _V
13	6, 12	eqeltri 2508	. . . . 5 |- B e. _V
14	13	mptex 5969	. . . 4 |- ( y e. B |-> ( X ( 1st ` F ) y ) ) e. _V
15	13, 13	mpt2ex 6428	. . . 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 6361	. . 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 16	. 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 453	. . 3 |- ( ( ph /\ y = Y ) -> Z e. B )
21		ovex 6109	. . . . 5 |- ( y J z ) e. _V
22	21	mptex 5969	. . . 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 453	. . . . 5 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> H e. ( Y J Z ) )
26		simprl 734	. . . . . 6 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> y = Y )
27		simprr 735	. . . . . 6 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> z = Z )
28	26, 27	oveq12d 6102	. . . . 5 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> ( y J z ) = ( Y J Z ) )
29	25, 28	eleqtrrd 2515	. . . 4 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> H e. ( y J z ) )
30		ovex 6109	. . . . 5 |- ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) e. _V
31	30	a1i 11	. . . 4 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> ( ( .1. ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) e. _V )
32		simplrl 738	. . . . . . 7 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> y = Y )
33	32	opeq2d 3993	. . . . . 6 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> <. X , y >. = <. X , Y >. )
34		simplrr 739	. . . . . . 7 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> z = Z )
35	34	opeq2d 3993	. . . . . 6 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> <. X , z >. = <. X , Z >. )
36	33, 35	oveq12d 6102	. . . . 5 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> ( <. X , y >. ( 2nd ` F ) <. X , z >. ) = ( <. X , Y >. ( 2nd ` F ) <. X , Z >. ) )
37		eqidd 2439	. . . . 5 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> ( .1. ` X ) = ( .1. ` X ) )
38		simpr 449	. . . . 5 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> g = H )
39	36, 37, 38	oveq123d 6105	. . . 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 ) )
40	29, 31, 39	fvmptdv2 5821	. . 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 ) ) )
41	18, 20, 23, 40	ovmpt2dv 6209	. 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 ) ) )
42	17, 41	mpd 15	1 |- ( ph -> ( ( Y ( 2nd ` K ) Z ) ` H ) = ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` F ) <. X , Z >. ) H ) )

Syntax hints:   -> wi 4   /\ wa 360   = wceq 1653   e. wcel 1726   _Vcvv 2958   <.cop 3819   e. cmpt 4269   ` cfv 5457  (class class class)co 6084   e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351   Basecbs 13474   Hom chom 13545   Catccat 13894   Idccid 13895   Func cfunc 14056   X._c cxpc 14270   curry_F ccurf 14312

This theorem is referenced by:  curf1cl  14330  curf2cl  14333  uncfcurf  14341  diag12  14346  yon12  14367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-curf 14316