Theorem oppchomfval 16374

Description: Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
oppchom.h	|- H = ( Hom ` C )
oppchom.o	|- O = (oppCat ` C )

Assertion

Ref	Expression
oppchomfval	|- tpos H = ( Hom ` O )

Proof of Theorem oppchomfval

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

Step	Hyp	Ref	Expression
1		homid 16075	. . . 4 |- Hom = Slot ( Hom ` ndx )
2		1nn0 11308	. . . . . . . 8 |- 1 e. NN0
3		4nn 11187	. . . . . . . 8 |- 4 e. NN
4	2, 3	decnncl 11518	. . . . . . 7 |- ; 1 4 e. NN
5	4	nnrei 11029	. . . . . 6 |- ; 1 4 e. RR
6		4nn0 11311	. . . . . . 7 |- 4 e. NN0
7		5nn 11188	. . . . . . 7 |- 5 e. NN
8		4lt5 11200	. . . . . . 7 |- 4 < 5
9	2, 6, 7, 8	declt 11530	. . . . . 6 |- ; 1 4 < ; 1 5
10	5, 9	ltneii 10150	. . . . 5 |- ; 1 4 =/= ; 1 5
11		homndx 16074	. . . . . 6 |- ( Hom ` ndx ) = ; 1 4
12		ccondx 16076	. . . . . 6 |- (comp ` ndx ) = ; 1 5
13	11, 12	neeq12i 2860	. . . . 5 |- ( ( Hom ` ndx ) =/= (comp ` ndx ) <-> ; 1 4 =/= ; 1 5 )
14	10, 13	mpbir 221	. . . 4 |- ( Hom ` ndx ) =/= (comp ` ndx )
15	1, 14	setsnid 15915	. . 3 |- ( Hom ` ( C sSet <. ( Hom ` ndx ) , tpos H >. ) ) = ( Hom ` ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( ( Base ` C ) X. ( Base ` C ) ) , z e. ( Base ` C ) |-> tpos ( <. z , ( 2nd ` u ) >. (comp ` C ) ( 1st ` u ) ) ) >. ) )
16		oppchom.h	. . . . . 6 |- H = ( Hom ` C )
17		fvex 6201	. . . . . 6 |- ( Hom ` C ) e. _V
18	16, 17	eqeltri 2697	. . . . 5 |- H e. _V
19	18	tposex 7386	. . . 4 |- tpos H e. _V
20	1	setsid 15914	. . . 4 |- ( ( C e. _V /\ tpos H e. _V ) -> tpos H = ( Hom ` ( C sSet <. ( Hom ` ndx ) , tpos H >. ) ) )
21	19, 20	mpan2 707	. . 3 |- ( C e. _V -> tpos H = ( Hom ` ( C sSet <. ( Hom ` ndx ) , tpos H >. ) ) )
22		eqid 2622	. . . . 5 |- ( Base ` C ) = ( Base ` C )
23		eqid 2622	. . . . 5 |- (comp ` C ) = (comp ` C )
24		oppchom.o	. . . . 5 |- O = (oppCat ` C )
25	22, 16, 23, 24	oppcval 16373	. . . 4 |- ( C e. _V -> O = ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( ( Base ` C ) X. ( Base ` C ) ) , z e. ( Base ` C ) |-> tpos ( <. z , ( 2nd ` u ) >. (comp ` C ) ( 1st ` u ) ) ) >. ) )
26	25	fveq2d 6195	. . 3 |- ( C e. _V -> ( Hom ` O ) = ( Hom ` ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( ( Base ` C ) X. ( Base ` C ) ) , z e. ( Base ` C ) |-> tpos ( <. z , ( 2nd ` u ) >. (comp ` C ) ( 1st ` u ) ) ) >. ) ) )
27	15, 21, 26	3eqtr4a 2682	. 2 |- ( C e. _V -> tpos H = ( Hom ` O ) )
28		tpos0 7382	. . 3 |- tpos (/) = (/)
29		fvprc 6185	. . . . 5 |- ( -. C e. _V -> ( Hom ` C ) = (/) )
30	16, 29	syl5eq 2668	. . . 4 |- ( -. C e. _V -> H = (/) )
31	30	tposeqd 7355	. . 3 |- ( -. C e. _V -> tpos H = tpos (/) )
32		fvprc 6185	. . . . . 6 |- ( -. C e. _V -> (oppCat ` C ) = (/) )
33	24, 32	syl5eq 2668	. . . . 5 |- ( -. C e. _V -> O = (/) )
34	33	fveq2d 6195	. . . 4 |- ( -. C e. _V -> ( Hom ` O ) = ( Hom ` (/) ) )
35		df-hom 15966	. . . . 5 |- Hom = Slot ; 1 4
36	35	str0 15911	. . . 4 |- (/) = ( Hom ` (/) )
37	34, 36	syl6eqr 2674	. . 3 |- ( -. C e. _V -> ( Hom ` O ) = (/) )
38	28, 31, 37	3eqtr4a 2682	. 2 |- ( -. C e. _V -> tpos H = ( Hom ` O ) )
39	27, 38	pm2.61i 176	1 |- tpos H = ( Hom ` O )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167  tpos ctpos 7351   1c1 9937   4c4 11072   5c5 11073  ;cdc 11493   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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  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-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-dec 11494  df-ndx 15860  df-slot 15861  df-sets 15864  df-hom 15966  df-cco 15967  df-oppc 16372

This theorem is referenced by:  oppchom  16375