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Theorem oppcbas 13872
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 2388 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2388 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2388 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
5 oppcbas.1 . . . . . 6  |-  O  =  (oppCat `  C )
62, 3, 4, 5oppcval 13867 . . . . 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 5673 . . . 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 13439 . . . . . 6  |-  Base  = Slot  ( Base `  ndx )
9 1re 9024 . . . . . . . 8  |-  1  e.  RR
10 1nn 9944 . . . . . . . . 9  |-  1  e.  NN
11 4nn0 10173 . . . . . . . . 9  |-  4  e.  NN0
12 1nn0 10170 . . . . . . . . 9  |-  1  e.  NN0
13 1lt10 10119 . . . . . . . . 9  |-  1  <  10
1410, 11, 12, 13declti 10340 . . . . . . . 8  |-  1  < ; 1
4
159, 14ltneii 9118 . . . . . . 7  |-  1  =/= ; 1 4
16 basendx 13442 . . . . . . . 8  |-  ( Base `  ndx )  =  1
17 homndx 13570 . . . . . . . 8  |-  (  Hom  `  ndx )  = ; 1 4
1816, 17neeq12i 2563 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (  Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
1915, 18mpbir 201 . . . . . 6  |-  ( Base `  ndx )  =/=  (  Hom  `  ndx )
208, 19setsnid 13437 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( C sSet  <.
(  Hom  `  ndx ) , tpos  (  Hom  `  C
) >. ) )
21 5nn 10069 . . . . . . . . . 10  |-  5  e.  NN
22 4lt5 10081 . . . . . . . . . 10  |-  4  <  5
2312, 11, 21, 22declt 10336 . . . . . . . . 9  |- ; 1 4  < ; 1 5
24 4nn 10068 . . . . . . . . . . . 12  |-  4  e.  NN
2512, 24decnncl 10328 . . . . . . . . . . 11  |- ; 1 4  e.  NN
2625nnrei 9942 . . . . . . . . . 10  |- ; 1 4  e.  RR
2712, 21decnncl 10328 . . . . . . . . . . 11  |- ; 1 5  e.  NN
2827nnrei 9942 . . . . . . . . . 10  |- ; 1 5  e.  RR
299, 26, 28lttri 9132 . . . . . . . . 9  |-  ( ( 1  < ; 1 4  /\ ; 1 4  < ; 1 5 )  -> 
1  < ; 1 5 )
3014, 23, 29mp2an 654 . . . . . . . 8  |-  1  < ; 1
5
319, 30ltneii 9118 . . . . . . 7  |-  1  =/= ; 1 5
32 ccondx 13572 . . . . . . . 8  |-  (comp `  ndx )  = ; 1 5
3316, 32neeq12i 2563 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (comp `  ndx )  <->  1  =/= ; 1 5 )
3431, 33mpbir 201 . . . . . 6  |-  ( Base `  ndx )  =/=  (comp ` 
ndx )
358, 34setsnid 13437 . . . . 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 2408 . . . 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 2439 . . 3  |-  ( C  e.  _V  ->  ( Base `  C )  =  ( Base `  O
) )
38 base0 13434 . . . 4  |-  (/)  =  (
Base `  (/) )
39 fvprc 5663 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
40 fvprc 5663 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
415, 40syl5eq 2432 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
4241fveq2d 5673 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  O )  =  ( Base `  (/) ) )
4338, 39, 423eqtr4a 2446 . . 3  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  ( Base `  O
) )
4437, 43pm2.61i 158 . 2  |-  ( Base `  C )  =  (
Base `  O )
451, 44eqtri 2408 1  |-  B  =  ( Base `  O
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900   (/)c0 3572   <.cop 3761   class class class wbr 4154    X. cxp 4817   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   1stc1st 6287   2ndc2nd 6288  tpos ctpos 6415   1c1 8925    < clt 9054   4c4 9984   5c5 9985  ;cdc 10315   ndxcnx 13394   sSet csts 13395   Basecbs 13397    Hom chom 13468  compcco 13469  oppCatcoppc 13865
This theorem is referenced by:  oppccatid  13873  oppchomf  13874  2oppcbas  13877  2oppccomf  13879  oppccomfpropd  13881  isepi  13894  epii  13897  oppcsect  13927  oppcsect2  13928  oppcinv  13929  oppciso  13930  sectepi  13933  episect  13934  funcoppc  14000  fulloppc  14047  fthoppc  14048  fthepi  14053  hofcl  14284  yon11  14289  yon12  14290  yon2  14291  oyon1cl  14296  yonedalem21  14298  yonedalem3a  14299  yonedalem4c  14302  yonedalem22  14303  yonedalem3b  14304  yonedalem3  14305  yonedainv  14306  yonffthlem  14307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-hom 13481  df-cco 13482  df-oppc 13866
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