Theorem oppcval 16373

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

Hypotheses

Ref	Expression
oppcval.b	|- B = ( Base ` C )
oppcval.h	|- H = ( Hom ` C )
oppcval.x	|- .x. = (comp ` C )
oppcval.o	|- O = (oppCat ` C )

Assertion

Ref	Expression
oppcval	|- ( C e. V -> O = ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) )

Distinct variable group:   z, u, C

Allowed substitution hints:   B( z, u)   .x. ( z, u)   H( z, u)   O( z, u)   V( z, u)

Proof of Theorem oppcval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppcval.o	. 2 |- O = (oppCat ` C )
2		elex 3212	. . 3 |- ( C e. V -> C e. _V )
3		id 22	. . . . . 6 |- ( c = C -> c = C )
4		fveq2 6191	. . . . . . . . 9 |- ( c = C -> ( Hom ` c ) = ( Hom ` C ) )
5		oppcval.h	. . . . . . . . 9 |- H = ( Hom ` C )
6	4, 5	syl6eqr 2674	. . . . . . . 8 |- ( c = C -> ( Hom ` c ) = H )
7	6	tposeqd 7355	. . . . . . 7 |- ( c = C -> tpos ( Hom ` c ) = tpos H )
8	7	opeq2d 4409	. . . . . 6 |- ( c = C -> <. ( Hom ` ndx ) , tpos ( Hom ` c ) >. = <. ( Hom ` ndx ) , tpos H >. )
9	3, 8	oveq12d 6668	. . . . 5 |- ( c = C -> ( c sSet <. ( Hom ` ndx ) , tpos ( Hom ` c ) >. ) = ( C sSet <. ( Hom ` ndx ) , tpos H >. ) )
10		fveq2 6191	. . . . . . . . 9 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
11		oppcval.b	. . . . . . . . 9 |- B = ( Base ` C )
12	10, 11	syl6eqr 2674	. . . . . . . 8 |- ( c = C -> ( Base ` c ) = B )
13	12	sqxpeqd 5141	. . . . . . 7 |- ( c = C -> ( ( Base ` c ) X. ( Base ` c ) ) = ( B X. B ) )
14		fveq2 6191	. . . . . . . . . 10 |- ( c = C -> (comp ` c ) = (comp ` C ) )
15		oppcval.x	. . . . . . . . . 10 |- .x. = (comp ` C )
16	14, 15	syl6eqr 2674	. . . . . . . . 9 |- ( c = C -> (comp ` c ) = .x. )
17	16	oveqd 6667	. . . . . . . 8 |- ( c = C -> ( <. z , ( 2nd ` u ) >. (comp ` c ) ( 1st ` u ) ) = ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) )
18	17	tposeqd 7355	. . . . . . 7 |- ( c = C -> tpos ( <. z , ( 2nd ` u ) >. (comp ` c ) ( 1st ` u ) ) = tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) )
19	13, 12, 18	mpt2eq123dv 6717	. . . . . 6 |- ( c = C -> ( u e. ( ( Base ` c ) X. ( Base ` c ) ) , z e. ( Base ` c ) |-> tpos ( <. z , ( 2nd ` u ) >. (comp ` c ) ( 1st ` u ) ) ) = ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) )
20	19	opeq2d 4409	. . . . 5 |- ( c = C -> <. (comp ` ndx ) , ( u e. ( ( Base ` c ) X. ( Base ` c ) ) , z e. ( Base ` c ) |-> tpos ( <. z , ( 2nd ` u ) >. (comp ` c ) ( 1st ` u ) ) ) >. = <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. )
21	9, 20	oveq12d 6668	. . . 4 |- ( c = C -> ( ( c sSet <. ( Hom ` ndx ) , tpos ( Hom ` c ) >. ) sSet <. (comp ` ndx ) , ( u e. ( ( Base ` c ) X. ( Base ` c ) ) , z e. ( Base ` c ) |-> tpos ( <. z , ( 2nd ` u ) >. (comp ` c ) ( 1st ` u ) ) ) >. ) = ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) )
22		df-oppc 16372	. . . 4 |- oppCat = ( c e. _V |-> ( ( c sSet <. ( Hom ` ndx ) , tpos ( Hom ` c ) >. ) sSet <. (comp ` ndx ) , ( u e. ( ( Base ` c ) X. ( Base ` c ) ) , z e. ( Base ` c ) |-> tpos ( <. z , ( 2nd ` u ) >. (comp ` c ) ( 1st ` u ) ) ) >. ) )
23		ovex 6678	. . . 4 |- ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) e. _V
24	21, 22, 23	fvmpt 6282	. . 3 |- ( C e. _V -> (oppCat ` C ) = ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) )
25	2, 24	syl 17	. 2 |- ( C e. V -> (oppCat ` C ) = ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) )
26	1, 25	syl5eq 2668	1 |- ( C e. V -> O = ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) )

Syntax hints:   -> wi 4   = 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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-tpos 7352  df-oppc 16372

This theorem is referenced by:  oppchomfval  16374  oppccofval  16376  oppcbas  16378  catcoppccl  16758