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Theorem oppcbas 15566
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 2428 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2428 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 eqid 2428 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
5 oppcbas.1 . . . . . 6  |-  O  =  (oppCat `  C )
62, 3, 4, 5oppcval 15561 . . . . 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 5829 . . . 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 15112 . . . . . 6  |-  Base  = Slot  ( Base `  ndx )
9 1re 9593 . . . . . . . 8  |-  1  e.  RR
10 1nn 10571 . . . . . . . . 9  |-  1  e.  NN
11 4nn0 10839 . . . . . . . . 9  |-  4  e.  NN0
12 1nn0 10836 . . . . . . . . 9  |-  1  e.  NN0
13 1lt10 10771 . . . . . . . . 9  |-  1  <  10
1410, 11, 12, 13declti 11027 . . . . . . . 8  |-  1  < ; 1
4
159, 14ltneii 9698 . . . . . . 7  |-  1  =/= ; 1 4
16 basendx 15116 . . . . . . . 8  |-  ( Base `  ndx )  =  1
17 homndx 15255 . . . . . . . 8  |-  ( Hom  `  ndx )  = ; 1 4
1816, 17neeq12i 2667 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  ( Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
1915, 18mpbir 212 . . . . . 6  |-  ( Base `  ndx )  =/=  ( Hom  `  ndx )
208, 19setsnid 15108 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( C sSet  <.
( Hom  `  ndx ) , tpos  ( Hom  `  C
) >. ) )
21 5nn 10721 . . . . . . . . . 10  |-  5  e.  NN
22 4lt5 10733 . . . . . . . . . 10  |-  4  <  5
2312, 11, 21, 22declt 11023 . . . . . . . . 9  |- ; 1 4  < ; 1 5
24 4nn 10720 . . . . . . . . . . . 12  |-  4  e.  NN
2512, 24decnncl 11015 . . . . . . . . . . 11  |- ; 1 4  e.  NN
2625nnrei 10569 . . . . . . . . . 10  |- ; 1 4  e.  RR
2712, 21decnncl 11015 . . . . . . . . . . 11  |- ; 1 5  e.  NN
2827nnrei 10569 . . . . . . . . . 10  |- ; 1 5  e.  RR
299, 26, 28lttri 9711 . . . . . . . . 9  |-  ( ( 1  < ; 1 4  /\ ; 1 4  < ; 1 5 )  -> 
1  < ; 1 5 )
3014, 23, 29mp2an 676 . . . . . . . 8  |-  1  < ; 1
5
319, 30ltneii 9698 . . . . . . 7  |-  1  =/= ; 1 5
32 ccondx 15257 . . . . . . . 8  |-  (comp `  ndx )  = ; 1 5
3316, 32neeq12i 2667 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (comp `  ndx )  <->  1  =/= ; 1 5 )
3431, 33mpbir 212 . . . . . 6  |-  ( Base `  ndx )  =/=  (comp ` 
ndx )
358, 34setsnid 15108 . . . . 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 2450 . . . 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 2481 . . 3  |-  ( C  e.  _V  ->  ( Base `  C )  =  ( Base `  O
) )
38 base0 15105 . . . 4  |-  (/)  =  (
Base `  (/) )
39 fvprc 5819 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
40 fvprc 5819 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
415, 40syl5eq 2474 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
4241fveq2d 5829 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  O )  =  ( Base `  (/) ) )
4338, 39, 423eqtr4a 2488 . . 3  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  ( Base `  O
) )
4437, 43pm2.61i 167 . 2  |-  ( Base `  C )  =  (
Base `  O )
451, 44eqtri 2450 1  |-  B  =  ( Base `  O
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437    e. wcel 1872    =/= wne 2599   _Vcvv 3022   (/)c0 3704   <.cop 3947   class class class wbr 4366    X. cxp 4794   ` cfv 5544  (class class class)co 6249    |-> cmpt2 6251   1stc1st 6749   2ndc2nd 6750  tpos ctpos 6927   1c1 9491    < clt 9626   4c4 10612   5c5 10613  ;cdc 11002   ndxcnx 15061   sSet csts 15062   Basecbs 15064   Hom chom 15144  compcco 15145  oppCatcoppc 15559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-tpos 6928  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-10 10627  df-n0 10821  df-z 10889  df-dec 11003  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-hom 15157  df-cco 15158  df-oppc 15560
This theorem is referenced by:  oppccatid  15567  oppchomf  15568  2oppcbas  15571  2oppccomf  15573  oppccomfpropd  15575  isepi  15588  epii  15591  oppcsect  15626  oppcsect2  15627  oppcinv  15628  oppciso  15629  sectepi  15632  episect  15633  funcoppc  15723  fulloppc  15770  fthoppc  15771  fthepi  15776  hofcl  16087  yon11  16092  yon12  16093  yon2  16094  oyon1cl  16099  yonedalem21  16101  yonedalem3a  16102  yonedalem4c  16105  yonedalem22  16106  yonedalem3b  16107  yonedalem3  16108  yonedainv  16109  yonffthlem  16110
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