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Theorem oppcbas 14649
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 2438 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2438 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 eqid 2438 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
5 oppcbas.1 . . . . . 6  |-  O  =  (oppCat `  C )
62, 3, 4, 5oppcval 14644 . . . . 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 5690 . . . 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 14212 . . . . . 6  |-  Base  = Slot  ( Base `  ndx )
9 1re 9377 . . . . . . . 8  |-  1  e.  RR
10 1nn 10325 . . . . . . . . 9  |-  1  e.  NN
11 4nn0 10590 . . . . . . . . 9  |-  4  e.  NN0
12 1nn0 10587 . . . . . . . . 9  |-  1  e.  NN0
13 1lt10 10524 . . . . . . . . 9  |-  1  <  10
1410, 11, 12, 13declti 10772 . . . . . . . 8  |-  1  < ; 1
4
159, 14ltneii 9479 . . . . . . 7  |-  1  =/= ; 1 4
16 basendx 14215 . . . . . . . 8  |-  ( Base `  ndx )  =  1
17 homndx 14345 . . . . . . . 8  |-  ( Hom  `  ndx )  = ; 1 4
1816, 17neeq12i 2615 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  ( Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
1915, 18mpbir 209 . . . . . 6  |-  ( Base `  ndx )  =/=  ( Hom  `  ndx )
208, 19setsnid 14208 . . . . 5  |-  ( Base `  C )  =  (
Base `  ( C sSet  <.
( Hom  `  ndx ) , tpos  ( Hom  `  C
) >. ) )
21 5nn 10474 . . . . . . . . . 10  |-  5  e.  NN
22 4lt5 10486 . . . . . . . . . 10  |-  4  <  5
2312, 11, 21, 22declt 10768 . . . . . . . . 9  |- ; 1 4  < ; 1 5
24 4nn 10473 . . . . . . . . . . . 12  |-  4  e.  NN
2512, 24decnncl 10760 . . . . . . . . . . 11  |- ; 1 4  e.  NN
2625nnrei 10323 . . . . . . . . . 10  |- ; 1 4  e.  RR
2712, 21decnncl 10760 . . . . . . . . . . 11  |- ; 1 5  e.  NN
2827nnrei 10323 . . . . . . . . . 10  |- ; 1 5  e.  RR
299, 26, 28lttri 9492 . . . . . . . . 9  |-  ( ( 1  < ; 1 4  /\ ; 1 4  < ; 1 5 )  -> 
1  < ; 1 5 )
3014, 23, 29mp2an 672 . . . . . . . 8  |-  1  < ; 1
5
319, 30ltneii 9479 . . . . . . 7  |-  1  =/= ; 1 5
32 ccondx 14347 . . . . . . . 8  |-  (comp `  ndx )  = ; 1 5
3316, 32neeq12i 2615 . . . . . . 7  |-  ( (
Base `  ndx )  =/=  (comp `  ndx )  <->  1  =/= ; 1 5 )
3431, 33mpbir 209 . . . . . 6  |-  ( Base `  ndx )  =/=  (comp ` 
ndx )
358, 34setsnid 14208 . . . . 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 2458 . . . 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 2489 . . 3  |-  ( C  e.  _V  ->  ( Base `  C )  =  ( Base `  O
) )
38 base0 14205 . . . 4  |-  (/)  =  (
Base `  (/) )
39 fvprc 5680 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
40 fvprc 5680 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
415, 40syl5eq 2482 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
4241fveq2d 5690 . . . 4  |-  ( -.  C  e.  _V  ->  (
Base `  O )  =  ( Base `  (/) ) )
4338, 39, 423eqtr4a 2496 . . 3  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  ( Base `  O
) )
4437, 43pm2.61i 164 . 2  |-  ( Base `  C )  =  (
Base `  O )
451, 44eqtri 2458 1  |-  B  =  ( Base `  O
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756    =/= wne 2601   _Vcvv 2967   (/)c0 3632   <.cop 3878   class class class wbr 4287    X. cxp 4833   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571  tpos ctpos 6739   1c1 9275    < clt 9410   4c4 10365   5c5 10366  ;cdc 10747   ndxcnx 14163   sSet csts 14164   Basecbs 14166   Hom chom 14241  compcco 14242  oppCatcoppc 14642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-tpos 6740  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-hom 14254  df-cco 14255  df-oppc 14643
This theorem is referenced by:  oppccatid  14650  oppchomf  14651  2oppcbas  14654  2oppccomf  14656  oppccomfpropd  14658  isepi  14671  epii  14674  oppcsect  14704  oppcsect2  14705  oppcinv  14706  oppciso  14707  sectepi  14710  episect  14711  funcoppc  14777  fulloppc  14824  fthoppc  14825  fthepi  14830  hofcl  15061  yon11  15066  yon12  15067  yon2  15068  oyon1cl  15073  yonedalem21  15075  yonedalem3a  15076  yonedalem4c  15079  yonedalem22  15080  yonedalem3b  15081  yonedalem3  15082  yonedainv  15083  yonffthlem  15084
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