MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oppcbas Structured version   Unicode version

Theorem oppcbas 13936
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 2435 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2435 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2435 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
5 oppcbas.1 . . . . . 6  |-  O  =  (oppCat `  C )
62, 3, 4, 5oppcval 13931 . . . . 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 5724 . . . 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 13503 . . . . . 6  |-  Base  = Slot  ( Base `  ndx )
9 1re 9082 . . . . . . . 8  |-  1  e.  RR
10 1nn 10003 . . . . . . . . 9  |-  1  e.  NN
11 4nn0 10232 . . . . . . . . 9  |-  4  e.  NN0
12 1nn0 10229 . . . . . . . . 9  |-  1  e.  NN0
13 1lt10 10178 . . . . . . . . 9  |-  1  <  10
1410, 11, 12, 13declti 10399 . . . . . . . 8  |-  1  < ; 1
4
159, 14ltneii 9178 . . . . . . 7  |-  1  =/= ; 1 4
16 basendx 13506 . . . . . . . 8  |-  ( Base `  ndx )  =  1
17 homndx 13634 . . . . . . . 8  |-  (  Hom  `  ndx )  = ; 1 4
1816, 17neeq12i 2610 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (  Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
1915, 18mpbir 201 . . . . . 6  |-  ( Base `  ndx )  =/=  (  Hom  `  ndx )
208, 19setsnid 13501 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( C sSet  <.
(  Hom  `  ndx ) , tpos  (  Hom  `  C
) >. ) )
21 5nn 10128 . . . . . . . . . 10  |-  5  e.  NN
22 4lt5 10140 . . . . . . . . . 10  |-  4  <  5
2312, 11, 21, 22declt 10395 . . . . . . . . 9  |- ; 1 4  < ; 1 5
24 4nn 10127 . . . . . . . . . . . 12  |-  4  e.  NN
2512, 24decnncl 10387 . . . . . . . . . . 11  |- ; 1 4  e.  NN
2625nnrei 10001 . . . . . . . . . 10  |- ; 1 4  e.  RR
2712, 21decnncl 10387 . . . . . . . . . . 11  |- ; 1 5  e.  NN
2827nnrei 10001 . . . . . . . . . 10  |- ; 1 5  e.  RR
299, 26, 28lttri 9191 . . . . . . . . 9  |-  ( ( 1  < ; 1 4  /\ ; 1 4  < ; 1 5 )  -> 
1  < ; 1 5 )
3014, 23, 29mp2an 654 . . . . . . . 8  |-  1  < ; 1
5
319, 30ltneii 9178 . . . . . . 7  |-  1  =/= ; 1 5
32 ccondx 13636 . . . . . . . 8  |-  (comp `  ndx )  = ; 1 5
3316, 32neeq12i 2610 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (comp `  ndx )  <->  1  =/= ; 1 5 )
3431, 33mpbir 201 . . . . . 6  |-  ( Base `  ndx )  =/=  (comp ` 
ndx )
358, 34setsnid 13501 . . . . 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 2455 . . . 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 2486 . . 3  |-  ( C  e.  _V  ->  ( Base `  C )  =  ( Base `  O
) )
38 base0 13498 . . . 4  |-  (/)  =  (
Base `  (/) )
39 fvprc 5714 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
40 fvprc 5714 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
415, 40syl5eq 2479 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
4241fveq2d 5724 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  O )  =  ( Base `  (/) ) )
4338, 39, 423eqtr4a 2493 . . 3  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  ( Base `  O
) )
4437, 43pm2.61i 158 . 2  |-  ( Base `  C )  =  (
Base `  O )
451, 44eqtri 2455 1  |-  B  =  ( Base `  O
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948   (/)c0 3620   <.cop 3809   class class class wbr 4204    X. cxp 4868   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340  tpos ctpos 6470   1c1 8983    < clt 9112   4c4 10043   5c5 10044  ;cdc 10374   ndxcnx 13458   sSet csts 13459   Basecbs 13461    Hom chom 13532  compcco 13533  oppCatcoppc 13929
This theorem is referenced by:  oppccatid  13937  oppchomf  13938  2oppcbas  13941  2oppccomf  13943  oppccomfpropd  13945  isepi  13958  epii  13961  oppcsect  13991  oppcsect2  13992  oppcinv  13993  oppciso  13994  sectepi  13997  episect  13998  funcoppc  14064  fulloppc  14111  fthoppc  14112  fthepi  14117  hofcl  14348  yon11  14353  yon12  14354  yon2  14355  oyon1cl  14360  yonedalem21  14362  yonedalem3a  14363  yonedalem4c  14366  yonedalem22  14367  yonedalem3b  14368  yonedalem3  14369  yonedainv  14370  yonffthlem  14371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-hom 13545  df-cco 13546  df-oppc 13930
  Copyright terms: Public domain W3C validator