Theorem uncf1 16876

Description: Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
uncfval.g	|- F = ( <" C D E "> uncurryF G )
uncfval.c	|- ( ph -> D e. Cat )
uncfval.d	|- ( ph -> E e. Cat )
uncfval.f	|- ( ph -> G e. ( C Func ( D FuncCat E ) ) )
uncf1.a	|- A = ( Base ` C )
uncf1.b	|- B = ( Base ` D )
uncf1.x	|- ( ph -> X e. A )
uncf1.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
uncf1	|- ( ph -> ( X ( 1st ` F ) Y ) = ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` Y ) )

Proof of Theorem uncf1

Step	Hyp	Ref	Expression
1		uncfval.g	. . . . 5 |- F = ( <" C D E "> uncurryF G )
2		uncfval.c	. . . . 5 |- ( ph -> D e. Cat )
3		uncfval.d	. . . . 5 |- ( ph -> E e. Cat )
4		uncfval.f	. . . . 5 |- ( ph -> G e. ( C Func ( D FuncCat E ) ) )
5	1, 2, 3, 4	uncfval 16874	. . . 4 |- ( ph -> F = ( ( D evalF E ) o.func ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) )
6	5	fveq2d 6195	. . 3 |- ( ph -> ( 1st ` F ) = ( 1st ` ( ( D evalF E ) o.func ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ) )
7	6	oveqd 6667	. 2 |- ( ph -> ( X ( 1st ` F ) Y ) = ( X ( 1st ` ( ( D evalF E ) o.func ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ) Y ) )
8		df-ov 6653	. . 3 |- ( X ( 1st ` ( ( D evalF E ) o.func ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ) Y ) = ( ( 1st ` ( ( D evalF E ) o.func ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ) ` <. X , Y >. )
9		eqid 2622	. . . . 5 |- ( C X.c D ) = ( C X.c D )
10		uncf1.a	. . . . 5 |- A = ( Base ` C )
11		uncf1.b	. . . . 5 |- B = ( Base ` D )
12	9, 10, 11	xpcbas 16818	. . . 4 |- ( A X. B ) = ( Base ` ( C X.c D ) )
13		eqid 2622	. . . . 5 |- ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) = ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) )
14		eqid 2622	. . . . 5 |- ( ( D FuncCat E ) X.c D ) = ( ( D FuncCat E ) X.c D )
15		funcrcl 16523	. . . . . . . . 9 |- ( G e. ( C Func ( D FuncCat E ) ) -> ( C e. Cat /\ ( D FuncCat E ) e. Cat ) )
16	4, 15	syl 17	. . . . . . . 8 |- ( ph -> ( C e. Cat /\ ( D FuncCat E ) e. Cat ) )
17	16	simpld 475	. . . . . . 7 |- ( ph -> C e. Cat )
18		eqid 2622	. . . . . . 7 |- ( C 1stF D ) = ( C 1stF D )
19	9, 17, 2, 18	1stfcl 16837	. . . . . 6 |- ( ph -> ( C 1stF D ) e. ( ( C X.c D ) Func C ) )
20	19, 4	cofucl 16548	. . . . 5 |- ( ph -> ( G o.func ( C 1stF D ) ) e. ( ( C X.c D ) Func ( D FuncCat E ) ) )
21		eqid 2622	. . . . . 6 |- ( C 2ndF D ) = ( C 2ndF D )
22	9, 17, 2, 21	2ndfcl 16838	. . . . 5 |- ( ph -> ( C 2ndF D ) e. ( ( C X.c D ) Func D ) )
23	13, 14, 20, 22	prfcl 16843	. . . 4 |- ( ph -> ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) e. ( ( C X.c D ) Func ( ( D FuncCat E ) X.c D ) ) )
24		eqid 2622	. . . . 5 |- ( D evalF E ) = ( D evalF E )
25		eqid 2622	. . . . 5 |- ( D FuncCat E ) = ( D FuncCat E )
26	24, 25, 2, 3	evlfcl 16862	. . . 4 |- ( ph -> ( D evalF E ) e. ( ( ( D FuncCat E ) X.c D ) Func E ) )
27		uncf1.x	. . . . 5 |- ( ph -> X e. A )
28		uncf1.y	. . . . 5 |- ( ph -> Y e. B )
29		opelxpi 5148	. . . . 5 |- ( ( X e. A /\ Y e. B ) -> <. X , Y >. e. ( A X. B ) )
30	27, 28, 29	syl2anc 693	. . . 4 |- ( ph -> <. X , Y >. e. ( A X. B ) )
31	12, 23, 26, 30	cofu1 16544	. . 3 |- ( ph -> ( ( 1st ` ( ( D evalF E ) o.func ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ) ` <. X , Y >. ) = ( ( 1st ` ( D evalF E ) ) ` ( ( 1st ` ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ` <. X , Y >. ) ) )
32	8, 31	syl5eq 2668	. 2 |- ( ph -> ( X ( 1st ` ( ( D evalF E ) o.func ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ) Y ) = ( ( 1st ` ( D evalF E ) ) ` ( ( 1st ` ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ` <. X , Y >. ) ) )
33		eqid 2622	. . . . . . 7 |- ( Hom ` ( C X.c D ) ) = ( Hom ` ( C X.c D ) )
34	13, 12, 33, 20, 22, 30	prf1 16840	. . . . . 6 |- ( ph -> ( ( 1st ` ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ` <. X , Y >. ) = <. ( ( 1st ` ( G o.func ( C 1stF D ) ) ) ` <. X , Y >. ) , ( ( 1st ` ( C 2ndF D ) ) ` <. X , Y >. ) >. )
35	12, 19, 4, 30	cofu1 16544	. . . . . . . 8 |- ( ph -> ( ( 1st ` ( G o.func ( C 1stF D ) ) ) ` <. X , Y >. ) = ( ( 1st ` G ) ` ( ( 1st ` ( C 1stF D ) ) ` <. X , Y >. ) ) )
36	9, 12, 33, 17, 2, 18, 30	1stf1 16832	. . . . . . . . . 10 |- ( ph -> ( ( 1st ` ( C 1stF D ) ) ` <. X , Y >. ) = ( 1st ` <. X , Y >. ) )
37		op1stg 7180	. . . . . . . . . . 11 |- ( ( X e. A /\ Y e. B ) -> ( 1st ` <. X , Y >. ) = X )
38	27, 28, 37	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( 1st ` <. X , Y >. ) = X )
39	36, 38	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( 1st ` ( C 1stF D ) ) ` <. X , Y >. ) = X )
40	39	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( ( 1st ` G ) ` ( ( 1st ` ( C 1stF D ) ) ` <. X , Y >. ) ) = ( ( 1st ` G ) ` X ) )
41	35, 40	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( 1st ` ( G o.func ( C 1stF D ) ) ) ` <. X , Y >. ) = ( ( 1st ` G ) ` X ) )
42	9, 12, 33, 17, 2, 21, 30	2ndf1 16835	. . . . . . . 8 |- ( ph -> ( ( 1st ` ( C 2ndF D ) ) ` <. X , Y >. ) = ( 2nd ` <. X , Y >. ) )
43		op2ndg 7181	. . . . . . . . 9 |- ( ( X e. A /\ Y e. B ) -> ( 2nd ` <. X , Y >. ) = Y )
44	27, 28, 43	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 2nd ` <. X , Y >. ) = Y )
45	42, 44	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( 1st ` ( C 2ndF D ) ) ` <. X , Y >. ) = Y )
46	41, 45	opeq12d 4410	. . . . . 6 |- ( ph -> <. ( ( 1st ` ( G o.func ( C 1stF D ) ) ) ` <. X , Y >. ) , ( ( 1st ` ( C 2ndF D ) ) ` <. X , Y >. ) >. = <. ( ( 1st ` G ) ` X ) , Y >. )
47	34, 46	eqtrd 2656	. . . . 5 |- ( ph -> ( ( 1st ` ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ` <. X , Y >. ) = <. ( ( 1st ` G ) ` X ) , Y >. )
48	47	fveq2d 6195	. . . 4 |- ( ph -> ( ( 1st ` ( D evalF E ) ) ` ( ( 1st ` ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ` <. X , Y >. ) ) = ( ( 1st ` ( D evalF E ) ) ` <. ( ( 1st ` G ) ` X ) , Y >. ) )
49		df-ov 6653	. . . 4 |- ( ( ( 1st ` G ) ` X ) ( 1st ` ( D evalF E ) ) Y ) = ( ( 1st ` ( D evalF E ) ) ` <. ( ( 1st ` G ) ` X ) , Y >. )
50	48, 49	syl6eqr 2674	. . 3 |- ( ph -> ( ( 1st ` ( D evalF E ) ) ` ( ( 1st ` ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ` <. X , Y >. ) ) = ( ( ( 1st ` G ) ` X ) ( 1st ` ( D evalF E ) ) Y ) )
51	25	fucbas 16620	. . . . . 6 |- ( D Func E ) = ( Base ` ( D FuncCat E ) )
52		relfunc 16522	. . . . . . 7 |- Rel ( C Func ( D FuncCat E ) )
53		1st2ndbr 7217	. . . . . . 7 |- ( ( Rel ( C Func ( D FuncCat E ) ) /\ G e. ( C Func ( D FuncCat E ) ) ) -> ( 1st ` G ) ( C Func ( D FuncCat E ) ) ( 2nd ` G ) )
54	52, 4, 53	sylancr 695	. . . . . 6 |- ( ph -> ( 1st ` G ) ( C Func ( D FuncCat E ) ) ( 2nd ` G ) )
55	10, 51, 54	funcf1 16526	. . . . 5 |- ( ph -> ( 1st ` G ) : A --> ( D Func E ) )
56	55, 27	ffvelrnd 6360	. . . 4 |- ( ph -> ( ( 1st ` G ) ` X ) e. ( D Func E ) )
57	24, 2, 3, 11, 56, 28	evlf1 16860	. . 3 |- ( ph -> ( ( ( 1st ` G ) ` X ) ( 1st ` ( D evalF E ) ) Y ) = ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` Y ) )
58	50, 57	eqtrd 2656	. 2 |- ( ph -> ( ( 1st ` ( D evalF E ) ) ` ( ( 1st ` ( ( G o.func ( C 1stF D ) ) ⟨,⟩F ( C 2ndF D ) ) ) ` <. X , Y >. ) ) = ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` Y ) )
59	7, 32, 58	3eqtrd 2660	1 |- ( ph -> ( X ( 1st ` F ) Y ) = ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` Y ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   X. cxp 5112   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   <"cs3 13587   Basecbs 15857   Hom chom 15952   Catccat 16325   Func cfunc 16514   o._func ccofu 16516   FuncCat cfuc 16602   X._c cxpc 16808   1st_F c1stf 16809   2nd_F c2ndf 16810   ⟨,⟩_F cprf 16811   eval_F cevlf 16849   uncurry_F cuncf 16851

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-func 16518  df-cofu 16520  df-nat 16603  df-fuc 16604  df-xpc 16812  df-1stf 16813  df-2ndf 16814  df-prf 16815  df-evlf 16853  df-uncf 16855

This theorem is referenced by:  curfuncf  16878  uncfcurf  16879