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Theorem oppcbas 14974
Description: Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
oppcbas.1  |-  O  =  (oppCat `  C )
oppcbas.2  |-  B  =  ( Base `  C
)
Assertion
Ref Expression
oppcbas  |-  B  =  ( Base `  O
)

Proof of Theorem oppcbas
Dummy variables  u  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppcbas.2 . 2  |-  B  =  ( Base `  C
)
2 eqid 2467 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2467 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 eqid 2467 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
5 oppcbas.1 . . . . . 6  |-  O  =  (oppCat `  C )
62, 3, 4, 5oppcval 14969 . . . . 5  |-  ( C  e.  _V  ->  O  =  ( ( 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 ) ) )
>. ) )
76fveq2d 5870 . . . 4  |-  ( C  e.  _V  ->  ( Base `  O )  =  ( Base `  (
( 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 ) ) )
>. ) ) )
8 baseid 14536 . . . . . 6  |-  Base  = Slot  ( Base `  ndx )
9 1re 9595 . . . . . . . 8  |-  1  e.  RR
10 1nn 10547 . . . . . . . . 9  |-  1  e.  NN
11 4nn0 10814 . . . . . . . . 9  |-  4  e.  NN0
12 1nn0 10811 . . . . . . . . 9  |-  1  e.  NN0
13 1lt10 10746 . . . . . . . . 9  |-  1  <  10
1410, 11, 12, 13declti 11001 . . . . . . . 8  |-  1  < ; 1
4
159, 14ltneii 9697 . . . . . . 7  |-  1  =/= ; 1 4
16 basendx 14540 . . . . . . . 8  |-  ( Base `  ndx )  =  1
17 homndx 14670 . . . . . . . 8  |-  ( Hom  `  ndx )  = ; 1 4
1816, 17neeq12i 2756 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  ( Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
1915, 18mpbir 209 . . . . . 6  |-  ( Base `  ndx )  =/=  ( Hom  `  ndx )
208, 19setsnid 14532 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( C sSet  <.
( Hom  `  ndx ) , tpos  ( Hom  `  C
) >. ) )
21 5nn 10696 . . . . . . . . . 10  |-  5  e.  NN
22 4lt5 10708 . . . . . . . . . 10  |-  4  <  5
2312, 11, 21, 22declt 10997 . . . . . . . . 9  |- ; 1 4  < ; 1 5
24 4nn 10695 . . . . . . . . . . . 12  |-  4  e.  NN
2512, 24decnncl 10989 . . . . . . . . . . 11  |- ; 1 4  e.  NN
2625nnrei 10545 . . . . . . . . . 10  |- ; 1 4  e.  RR
2712, 21decnncl 10989 . . . . . . . . . . 11  |- ; 1 5  e.  NN
2827nnrei 10545 . . . . . . . . . 10  |- ; 1 5  e.  RR
299, 26, 28lttri 9710 . . . . . . . . 9  |-  ( ( 1  < ; 1 4  /\ ; 1 4  < ; 1 5 )  -> 
1  < ; 1 5 )
3014, 23, 29mp2an 672 . . . . . . . 8  |-  1  < ; 1
5
319, 30ltneii 9697 . . . . . . 7  |-  1  =/= ; 1 5
32 ccondx 14672 . . . . . . . 8  |-  (comp `  ndx )  = ; 1 5
3316, 32neeq12i 2756 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (comp `  ndx )  <->  1  =/= ; 1 5 )
3431, 33mpbir 209 . . . . . 6  |-  ( Base `  ndx )  =/=  (comp ` 
ndx )
358, 34setsnid 14532 . . . . 5  |-  ( Base `  ( C sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  C
) >. ) )  =  ( Base `  (
( 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 ) ) )
>. ) )
3620, 35eqtri 2496 . . . 4  |-  ( Base `  C )  =  (
Base `  ( ( 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 ) ) )
>. ) )
377, 36syl6reqr 2527 . . 3  |-  ( C  e.  _V  ->  ( Base `  C )  =  ( Base `  O
) )
38 base0 14529 . . . 4  |-  (/)  =  (
Base `  (/) )
39 fvprc 5860 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
40 fvprc 5860 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
415, 40syl5eq 2520 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
4241fveq2d 5870 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  O )  =  ( Base `  (/) ) )
4338, 39, 423eqtr4a 2534 . . 3  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  ( Base `  O
) )
4437, 43pm2.61i 164 . 2  |-  ( Base `  C )  =  (
Base `  O )
451, 44eqtri 2496 1  |-  B  =  ( Base `  O
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   <.cop 4033   class class class wbr 4447    X. cxp 4997   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1stc1st 6782   2ndc2nd 6783  tpos ctpos 6954   1c1 9493    < clt 9628   4c4 10587   5c5 10588  ;cdc 10976   ndxcnx 14487   sSet csts 14488   Basecbs 14490   Hom chom 14566  compcco 14567  oppCatcoppc 14967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-hom 14579  df-cco 14580  df-oppc 14968
This theorem is referenced by:  oppccatid  14975  oppchomf  14976  2oppcbas  14979  2oppccomf  14981  oppccomfpropd  14983  isepi  14996  epii  14999  oppcsect  15029  oppcsect2  15030  oppcinv  15031  oppciso  15032  sectepi  15035  episect  15036  funcoppc  15102  fulloppc  15149  fthoppc  15150  fthepi  15155  hofcl  15386  yon11  15391  yon12  15392  yon2  15393  oyon1cl  15398  yonedalem21  15400  yonedalem3a  15401  yonedalem4c  15404  yonedalem22  15405  yonedalem3b  15406  yonedalem3  15407  yonedainv  15408  yonffthlem  15409
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