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Theorem oppchomfval 13867
Description: Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
oppchom.h  |-  H  =  (  Hom  `  C
)
oppchom.o  |-  O  =  (oppCat `  C )
Assertion
Ref Expression
oppchomfval  |- tpos  H  =  (  Hom  `  O
)

Proof of Theorem oppchomfval
Dummy variables  z  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 homid 13570 . . . 4  |-  Hom  = Slot  (  Hom  `  ndx )
2 1nn0 10169 . . . . . . . 8  |-  1  e.  NN0
3 4nn 10067 . . . . . . . 8  |-  4  e.  NN
42, 3decnncl 10327 . . . . . . 7  |- ; 1 4  e.  NN
54nnrei 9941 . . . . . 6  |- ; 1 4  e.  RR
6 4nn0 10172 . . . . . . 7  |-  4  e.  NN0
7 5nn 10068 . . . . . . 7  |-  5  e.  NN
8 4lt5 10080 . . . . . . 7  |-  4  <  5
92, 6, 7, 8declt 10335 . . . . . 6  |- ; 1 4  < ; 1 5
105, 9ltneii 9117 . . . . 5  |- ; 1 4  =/= ; 1 5
11 homndx 13569 . . . . . 6  |-  (  Hom  `  ndx )  = ; 1 4
12 ccondx 13571 . . . . . 6  |-  (comp `  ndx )  = ; 1 5
1311, 12neeq12i 2562 . . . . 5  |-  ( (  Hom  `  ndx )  =/=  (comp `  ndx )  <-> ; 1 4  =/= ; 1 5 )
1410, 13mpbir 201 . . . 4  |-  (  Hom  `  ndx )  =/=  (comp ` 
ndx )
151, 14setsnid 13436 . . 3  |-  (  Hom  `  ( C sSet  <. (  Hom  `  ndx ) , tpos 
H >. ) )  =  (  Hom  `  (
( C sSet  <. (  Hom  `  ndx ) , tpos  H >. ) sSet  <. (comp `  ndx ) ,  ( u  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ,  z  e.  ( Base `  C )  |-> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  C ) ( 1st `  u ) ) )
>. ) )
16 oppchom.h . . . . . 6  |-  H  =  (  Hom  `  C
)
17 fvex 5682 . . . . . 6  |-  (  Hom  `  C )  e.  _V
1816, 17eqeltri 2457 . . . . 5  |-  H  e. 
_V
1918tposex 6449 . . . 4  |- tpos  H  e. 
_V
201setsid 13435 . . . 4  |-  ( ( C  e.  _V  /\ tpos  H  e.  _V )  -> tpos  H  =  (  Hom  `  ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. )
) )
2119, 20mpan2 653 . . 3  |-  ( C  e.  _V  -> tpos  H  =  (  Hom  `  ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. )
) )
22 eqid 2387 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
23 eqid 2387 . . . . 5  |-  (comp `  C )  =  (comp `  C )
24 oppchom.o . . . . 5  |-  O  =  (oppCat `  C )
2522, 16, 23, 24oppcval 13866 . . . 4  |-  ( C  e.  _V  ->  O  =  ( ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. ) sSet  <.
(comp `  ndx ) ,  ( u  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ,  z  e.  ( Base `  C )  |-> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  C ) ( 1st `  u ) ) )
>. ) )
2625fveq2d 5672 . . 3  |-  ( C  e.  _V  ->  (  Hom  `  O )  =  (  Hom  `  (
( C sSet  <. (  Hom  `  ndx ) , tpos  H >. ) sSet  <. (comp `  ndx ) ,  ( u  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ,  z  e.  ( Base `  C )  |-> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  C ) ( 1st `  u ) ) )
>. ) ) )
2715, 21, 263eqtr4a 2445 . 2  |-  ( C  e.  _V  -> tpos  H  =  (  Hom  `  O
) )
28 tpos0 6445 . . 3  |- tpos  (/)  =  (/)
29 fvprc 5662 . . . . 5  |-  ( -.  C  e.  _V  ->  (  Hom  `  C )  =  (/) )
3016, 29syl5eq 2431 . . . 4  |-  ( -.  C  e.  _V  ->  H  =  (/) )
3130tposeqd 6418 . . 3  |-  ( -.  C  e.  _V  -> tpos  H  = tpos  (/) )
32 fvprc 5662 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
3324, 32syl5eq 2431 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
3433fveq2d 5672 . . . 4  |-  ( -.  C  e.  _V  ->  (  Hom  `  O )  =  (  Hom  `  (/) ) )
35 df-hom 13480 . . . . 5  |-  Hom  = Slot ; 1 4
3635str0 13432 . . . 4  |-  (/)  =  (  Hom  `  (/) )
3734, 36syl6eqr 2437 . . 3  |-  ( -.  C  e.  _V  ->  (  Hom  `  O )  =  (/) )
3828, 31, 373eqtr4a 2445 . 2  |-  ( -.  C  e.  _V  -> tpos  H  =  (  Hom  `  O
) )
3927, 38pm2.61i 158 1  |- tpos  H  =  (  Hom  `  O
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899   (/)c0 3571   <.cop 3760    X. cxp 4816   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1stc1st 6286   2ndc2nd 6287  tpos ctpos 6414   1c1 8924   4c4 9983   5c5 9984  ;cdc 10314   ndxcnx 13393   sSet csts 13394   Basecbs 13396    Hom chom 13467  compcco 13468  oppCatcoppc 13864
This theorem is referenced by:  oppchom  13868
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-dec 10315  df-ndx 13399  df-slot 13400  df-sets 13402  df-hom 13480  df-cco 13481  df-oppc 13865
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