Theorem oppccofval 16376

Description: Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
oppcco.b	|- B = ( Base ` C )
oppcco.c	|- .x. = (comp ` C )
oppcco.o	|- O = (oppCat ` C )
oppcco.x	|- ( ph -> X e. B )
oppcco.y	|- ( ph -> Y e. B )
oppcco.z	|- ( ph -> Z e. B )

Assertion

Ref	Expression
oppccofval	|- ( ph -> ( <. X , Y >. (comp ` O ) Z ) = tpos ( <. Z , Y >. .x. X ) )

Proof of Theorem oppccofval

Dummy variables z u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppcco.x	. . . . 5 |- ( ph -> X e. B )
2		elfvex 6221	. . . . . 6 |- ( X e. ( Base ` C ) -> C e. _V )
3		oppcco.b	. . . . . 6 |- B = ( Base ` C )
4	2, 3	eleq2s 2719	. . . . 5 |- ( X e. B -> C e. _V )
5		eqid 2622	. . . . . 6 |- ( Hom ` C ) = ( Hom ` C )
6		oppcco.c	. . . . . 6 |- .x. = (comp ` C )
7		oppcco.o	. . . . . 6 |- O = (oppCat ` C )
8	3, 5, 6, 7	oppcval 16373	. . . . 5 |- ( C e. _V -> O = ( ( C sSet <. ( Hom ` ndx ) , tpos ( Hom ` C ) >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) )
9	1, 4, 8	3syl 18	. . . 4 |- ( ph -> O = ( ( C sSet <. ( Hom ` ndx ) , tpos ( Hom ` C ) >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) )
10	9	fveq2d 6195	. . 3 |- ( ph -> (comp ` O ) = (comp ` ( ( C sSet <. ( Hom ` ndx ) , tpos ( Hom ` C ) >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) ) )
11		ovex 6678	. . . 4 |- ( C sSet <. ( Hom ` ndx ) , tpos ( Hom ` C ) >. ) e. _V
12		fvex 6201	. . . . . . 7 |- ( Base ` C ) e. _V
13	3, 12	eqeltri 2697	. . . . . 6 |- B e. _V
14	13, 13	xpex 6962	. . . . 5 |- ( B X. B ) e. _V
15	14, 13	mpt2ex 7247	. . . 4 |- ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) e. _V
16		ccoid 16077	. . . . 5 |- comp = Slot (comp ` ndx )
17	16	setsid 15914	. . . 4 |- ( ( ( C sSet <. ( Hom ` ndx ) , tpos ( Hom ` C ) >. ) e. _V /\ ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) e. _V ) -> ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) = (comp ` ( ( C sSet <. ( Hom ` ndx ) , tpos ( Hom ` C ) >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) ) )
18	11, 15, 17	mp2an 708	. . 3 |- ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) = (comp ` ( ( C sSet <. ( Hom ` ndx ) , tpos ( Hom ` C ) >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) )
19	10, 18	syl6eqr 2674	. 2 |- ( ph -> (comp ` O ) = ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) )
20		simprr 796	. . . . 5 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> z = Z )
21		simprl 794	. . . . . . 7 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> u = <. X , Y >. )
22	21	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` u ) = ( 2nd ` <. X , Y >. ) )
23	1	adantr 481	. . . . . . 7 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> X e. B )
24		oppcco.y	. . . . . . . 8 |- ( ph -> Y e. B )
25	24	adantr 481	. . . . . . 7 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> Y e. B )
26		op2ndg 7181	. . . . . . 7 |- ( ( X e. B /\ Y e. B ) -> ( 2nd ` <. X , Y >. ) = Y )
27	23, 25, 26	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` <. X , Y >. ) = Y )
28	22, 27	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` u ) = Y )
29	20, 28	opeq12d 4410	. . . 4 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> <. z , ( 2nd ` u ) >. = <. Z , Y >. )
30	21	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> ( 1st ` u ) = ( 1st ` <. X , Y >. ) )
31		op1stg 7180	. . . . . 6 |- ( ( X e. B /\ Y e. B ) -> ( 1st ` <. X , Y >. ) = X )
32	23, 25, 31	syl2anc 693	. . . . 5 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> ( 1st ` <. X , Y >. ) = X )
33	30, 32	eqtrd 2656	. . . 4 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> ( 1st ` u ) = X )
34	29, 33	oveq12d 6668	. . 3 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) = ( <. Z , Y >. .x. X ) )
35	34	tposeqd 7355	. 2 |- ( ( ph /\ ( u = <. X , Y >. /\ z = Z ) ) -> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) = tpos ( <. Z , Y >. .x. X ) )
36		opelxpi 5148	. . 3 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
37	1, 24, 36	syl2anc 693	. 2 |- ( ph -> <. X , Y >. e. ( B X. B ) )
38		oppcco.z	. 2 |- ( ph -> Z e. B )
39		ovex 6678	. . . 4 |- ( <. Z , Y >. .x. X ) e. _V
40	39	tposex 7386	. . 3 |- tpos ( <. Z , Y >. .x. X ) e. _V
41	40	a1i 11	. 2 |- ( ph -> tpos ( <. Z , Y >. .x. X ) e. _V )
42	19, 35, 37, 38, 41	ovmpt2d 6788	1 |- ( ph -> ( <. X , Y >. (comp ` O ) Z ) = tpos ( <. Z , Y >. .x. X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167  tpos ctpos 7351   ndxcnx 15854   sSet csts 15855   Basecbs 15857   Hom chom 15952  compcco 15953  oppCatcoppc 16371

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  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-dec 11494  df-ndx 15860  df-slot 15861  df-sets 15864  df-cco 15967  df-oppc 16372

This theorem is referenced by:  oppcco  16377