Theorem curf12 14287

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 14285	. . 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 5709	. . . . . 6 |- ( Base ` D ) e. _V
13	6, 12	eqeltri 2482	. . . . 5 |- B e. _V
14	13	mptex 5933	. . . 4 |- ( y e. B |-> ( X ( 1st ` F ) y ) ) e. _V
15	13, 13	mpt2ex 6392	. . . 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 6325	. . 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 452	. . 3 |- ( ( ph /\ y = Y ) -> Z e. B )
21		ovex 6073	. . . . 5 |- ( y J z ) e. _V
22	21	mptex 5933	. . . 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 452	. . . . 5 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> H e. ( Y J Z ) )
26		simprl 733	. . . . . 6 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> y = Y )
27		simprr 734	. . . . . 6 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> z = Z )
28	26, 27	oveq12d 6066	. . . . 5 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> ( y J z ) = ( Y J Z ) )
29	25, 28	eleqtrrd 2489	. . . 4 |- ( ( ph /\ ( y = Y /\ z = Z ) ) -> H e. ( y J z ) )
30		ovex 6073	. . . . 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 737	. . . . . . 7 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> y = Y )
33	32	opeq2d 3959	. . . . . 6 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> <. X , y >. = <. X , Y >. )
34		simplrr 738	. . . . . . 7 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> z = Z )
35	34	opeq2d 3959	. . . . . 6 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> <. X , z >. = <. X , Z >. )
36	33, 35	oveq12d 6066	. . . . 5 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> ( <. X , y >. ( 2nd ` F ) <. X , z >. ) = ( <. X , Y >. ( 2nd ` F ) <. X , Z >. ) )
37		eqidd 2413	. . . . 5 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> ( .1. ` X ) = ( .1. ` X ) )
38		simpr 448	. . . . 5 |- ( ( ( ph /\ ( y = Y /\ z = Z ) ) /\ g = H ) -> g = H )
39	36, 37, 38	oveq123d 6069	. . . 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 5785	. . 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 6173	. 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 359   = wceq 1649   e. wcel 1721   _Vcvv 2924   <.cop 3785   e. cmpt 4234   ` cfv 5421  (class class class)co 6048   e. cmpt2 6050   1stc1st 6314   2ndc2nd 6315   Basecbs 13432   Hom chom 13503   Catccat 13852   Idccid 13853   Func cfunc 14014   X._c cxpc 14228   curry_F ccurf 14270

This theorem is referenced by:  curf1cl  14288  curf2cl  14291  uncfcurf  14299  diag12  14304  yon12  14325

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-curf 14274