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Theorem oppcbas 13637
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 2296 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2296 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2296 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
5 oppcbas.1 . . . . . 6  |-  O  =  (oppCat `  C )
62, 3, 4, 5oppcval 13632 . . . . 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 5545 . . . 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 df-base 13169 . . . . . . 7  |-  Base  = Slot  1
9 1nn 9773 . . . . . . 7  |-  1  e.  NN
108, 9ndxid 13185 . . . . . 6  |-  Base  = Slot  ( Base `  ndx )
119nnrei 9771 . . . . . . . 8  |-  1  e.  RR
12 4nn0 10000 . . . . . . . . 9  |-  4  e.  NN0
13 1nn0 9997 . . . . . . . . 9  |-  1  e.  NN0
14 1lt10 9946 . . . . . . . . 9  |-  1  <  10
159, 12, 13, 14declti 10165 . . . . . . . 8  |-  1  < ; 1
4
1611, 15ltneii 8947 . . . . . . 7  |-  1  =/= ; 1 4
17 basendx 13209 . . . . . . . 8  |-  ( Base `  ndx )  =  1
18 df-hom 13248 . . . . . . . . 9  |-  Hom  = Slot ; 1 4
19 4nn 9895 . . . . . . . . . 10  |-  4  e.  NN
2013, 19decnncl 10153 . . . . . . . . 9  |- ; 1 4  e.  NN
2118, 20ndxarg 13184 . . . . . . . 8  |-  (  Hom  `  ndx )  = ; 1 4
2217, 21neeq12i 2471 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (  Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
2316, 22mpbir 200 . . . . . 6  |-  ( Base `  ndx )  =/=  (  Hom  `  ndx )
2410, 23setsnid 13204 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( C sSet  <.
(  Hom  `  ndx ) , tpos  (  Hom  `  C
) >. ) )
25 5nn 9896 . . . . . . . . . . 11  |-  5  e.  NN
26 4lt5 9908 . . . . . . . . . . 11  |-  4  <  5
2713, 12, 25, 26declt 10161 . . . . . . . . . 10  |- ; 1 4  < ; 1 5
2820nnrei 9771 . . . . . . . . . . 11  |- ; 1 4  e.  RR
2913, 25decnncl 10153 . . . . . . . . . . . 12  |- ; 1 5  e.  NN
3029nnrei 9771 . . . . . . . . . . 11  |- ; 1 5  e.  RR
3111, 28, 30lttri 8961 . . . . . . . . . 10  |-  ( ( 1  < ; 1 4  /\ ; 1 4  < ; 1 5 )  -> 
1  < ; 1 5 )
3227, 31mpan2 652 . . . . . . . . 9  |-  ( 1  < ; 1 4  ->  1  < ; 1
5 )
3315, 32ax-mp 8 . . . . . . . 8  |-  1  < ; 1
5
3411, 33ltneii 8947 . . . . . . 7  |-  1  =/= ; 1 5
35 df-cco 13249 . . . . . . . . 9  |- comp  = Slot ; 1 5
3635, 29ndxarg 13184 . . . . . . . 8  |-  (comp `  ndx )  = ; 1 5
3717, 36neeq12i 2471 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (comp `  ndx )  <->  1  =/= ; 1 5 )
3834, 37mpbir 200 . . . . . 6  |-  ( Base `  ndx )  =/=  (comp ` 
ndx )
3910, 38setsnid 13204 . . . . 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 ) ) )
>. ) )
4024, 39eqtri 2316 . . . 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
) ) ) >.
) )
417, 40syl6reqr 2347 . . 3  |-  ( C  e.  _V  ->  ( Base `  C )  =  ( Base `  O
) )
42 fvprc 5535 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
43 fvprc 5535 . . . . . . 7  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
445, 43syl5eq 2340 . . . . . 6  |-  ( -.  C  e.  _V  ->  O  =  (/) )
4544fveq2d 5545 . . . . 5  |-  ( -.  C  e.  _V  ->  (
Base `  O )  =  ( Base `  (/) ) )
468str0 13200 . . . . 5  |-  (/)  =  (
Base `  (/) )
4745, 46syl6eqr 2346 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  O )  =  (/) )
4842, 47eqtr4d 2331 . . 3  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  ( Base `  O
) )
4941, 48pm2.61i 156 . 2  |-  ( Base `  C )  =  (
Base `  O )
501, 49eqtri 2316 1  |-  B  =  ( Base `  O
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   <.cop 3656   class class class wbr 4039    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137  tpos ctpos 6249   1c1 8754    < clt 8883   4c4 9813   5c5 9814  ;cdc 10140   ndxcnx 13161   sSet csts 13162   Basecbs 13164    Hom chom 13235  compcco 13236  oppCatcoppc 13630
This theorem is referenced by:  oppccatid  13638  oppchomf  13639  2oppcbas  13642  2oppccomf  13644  oppccomfpropd  13646  isepi  13659  epii  13662  oppcsect  13692  oppcsect2  13693  oppcinv  13694  oppciso  13695  sectepi  13698  episect  13699  funcoppc  13765  fulloppc  13812  fthoppc  13813  fthepi  13818  hofcl  14049  yon11  14054  yon12  14055  yon2  14056  oyon1cl  14061  yonedalem21  14063  yonedalem3a  14064  yonedalem4c  14067  yonedalem22  14068  yonedalem3b  14069  yonedalem3  14070  yonedainv  14071  yonffthlem  14072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-hom 13248  df-cco 13249  df-oppc 13631
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