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Theorem oppcbas 14640
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 2433 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2433 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 eqid 2433 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
5 oppcbas.1 . . . . . 6  |-  O  =  (oppCat `  C )
62, 3, 4, 5oppcval 14635 . . . . 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 5683 . . . 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 14203 . . . . . 6  |-  Base  = Slot  ( Base `  ndx )
9 1re 9373 . . . . . . . 8  |-  1  e.  RR
10 1nn 10321 . . . . . . . . 9  |-  1  e.  NN
11 4nn0 10586 . . . . . . . . 9  |-  4  e.  NN0
12 1nn0 10583 . . . . . . . . 9  |-  1  e.  NN0
13 1lt10 10520 . . . . . . . . 9  |-  1  <  10
1410, 11, 12, 13declti 10768 . . . . . . . 8  |-  1  < ; 1
4
159, 14ltneii 9475 . . . . . . 7  |-  1  =/= ; 1 4
16 basendx 14206 . . . . . . . 8  |-  ( Base `  ndx )  =  1
17 homndx 14336 . . . . . . . 8  |-  ( Hom  `  ndx )  = ; 1 4
1816, 17neeq12i 2610 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  ( Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
1915, 18mpbir 209 . . . . . 6  |-  ( Base `  ndx )  =/=  ( Hom  `  ndx )
208, 19setsnid 14199 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( C sSet  <.
( Hom  `  ndx ) , tpos  ( Hom  `  C
) >. ) )
21 5nn 10470 . . . . . . . . . 10  |-  5  e.  NN
22 4lt5 10482 . . . . . . . . . 10  |-  4  <  5
2312, 11, 21, 22declt 10764 . . . . . . . . 9  |- ; 1 4  < ; 1 5
24 4nn 10469 . . . . . . . . . . . 12  |-  4  e.  NN
2512, 24decnncl 10756 . . . . . . . . . . 11  |- ; 1 4  e.  NN
2625nnrei 10319 . . . . . . . . . 10  |- ; 1 4  e.  RR
2712, 21decnncl 10756 . . . . . . . . . . 11  |- ; 1 5  e.  NN
2827nnrei 10319 . . . . . . . . . 10  |- ; 1 5  e.  RR
299, 26, 28lttri 9488 . . . . . . . . 9  |-  ( ( 1  < ; 1 4  /\ ; 1 4  < ; 1 5 )  -> 
1  < ; 1 5 )
3014, 23, 29mp2an 665 . . . . . . . 8  |-  1  < ; 1
5
319, 30ltneii 9475 . . . . . . 7  |-  1  =/= ; 1 5
32 ccondx 14338 . . . . . . . 8  |-  (comp `  ndx )  = ; 1 5
3316, 32neeq12i 2610 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (comp `  ndx )  <->  1  =/= ; 1 5 )
3431, 33mpbir 209 . . . . . 6  |-  ( Base `  ndx )  =/=  (comp ` 
ndx )
358, 34setsnid 14199 . . . . 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 2453 . . . 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 2484 . . 3  |-  ( C  e.  _V  ->  ( Base `  C )  =  ( Base `  O
) )
38 base0 14196 . . . 4  |-  (/)  =  (
Base `  (/) )
39 fvprc 5673 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
40 fvprc 5673 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
415, 40syl5eq 2477 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
4241fveq2d 5683 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  O )  =  ( Base `  (/) ) )
4338, 39, 423eqtr4a 2491 . . 3  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  ( Base `  O
) )
4437, 43pm2.61i 164 . 2  |-  ( Base `  C )  =  (
Base `  O )
451, 44eqtri 2453 1  |-  B  =  ( Base `  O
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1362    e. wcel 1755    =/= wne 2596   _Vcvv 2962   (/)c0 3625   <.cop 3871   class class class wbr 4280    X. cxp 4825   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   1stc1st 6564   2ndc2nd 6565  tpos ctpos 6733   1c1 9271    < clt 9406   4c4 10361   5c5 10362  ;cdc 10743   ndxcnx 14154   sSet csts 14155   Basecbs 14157   Hom chom 14232  compcco 14233  oppCatcoppc 14633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-tpos 6734  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-hom 14245  df-cco 14246  df-oppc 14634
This theorem is referenced by:  oppccatid  14641  oppchomf  14642  2oppcbas  14645  2oppccomf  14647  oppccomfpropd  14649  isepi  14662  epii  14665  oppcsect  14695  oppcsect2  14696  oppcinv  14697  oppciso  14698  sectepi  14701  episect  14702  funcoppc  14768  fulloppc  14815  fthoppc  14816  fthepi  14821  hofcl  15052  yon11  15057  yon12  15058  yon2  15059  oyon1cl  15064  yonedalem21  15066  yonedalem3a  15067  yonedalem4c  15070  yonedalem22  15071  yonedalem3b  15072  yonedalem3  15073  yonedainv  15074  yonffthlem  15075
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