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Theorem oppchomfval 13932
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 13635 . . . 4  |-  Hom  = Slot  (  Hom  `  ndx )
2 1nn0 10229 . . . . . . . 8  |-  1  e.  NN0
3 4nn 10127 . . . . . . . 8  |-  4  e.  NN
42, 3decnncl 10387 . . . . . . 7  |- ; 1 4  e.  NN
54nnrei 10001 . . . . . 6  |- ; 1 4  e.  RR
6 4nn0 10232 . . . . . . 7  |-  4  e.  NN0
7 5nn 10128 . . . . . . 7  |-  5  e.  NN
8 4lt5 10140 . . . . . . 7  |-  4  <  5
92, 6, 7, 8declt 10395 . . . . . 6  |- ; 1 4  < ; 1 5
105, 9ltneii 9178 . . . . 5  |- ; 1 4  =/= ; 1 5
11 homndx 13634 . . . . . 6  |-  (  Hom  `  ndx )  = ; 1 4
12 ccondx 13636 . . . . . 6  |-  (comp `  ndx )  = ; 1 5
1311, 12neeq12i 2610 . . . . 5  |-  ( (  Hom  `  ndx )  =/=  (comp `  ndx )  <-> ; 1 4  =/= ; 1 5 )
1410, 13mpbir 201 . . . 4  |-  (  Hom  `  ndx )  =/=  (comp ` 
ndx )
151, 14setsnid 13501 . . 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 5734 . . . . . 6  |-  (  Hom  `  C )  e.  _V
1816, 17eqeltri 2505 . . . . 5  |-  H  e. 
_V
1918tposex 6505 . . . 4  |- tpos  H  e. 
_V
201setsid 13500 . . . 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 2435 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
23 eqid 2435 . . . . 5  |-  (comp `  C )  =  (comp `  C )
24 oppchom.o . . . . 5  |-  O  =  (oppCat `  C )
2522, 16, 23, 24oppcval 13931 . . . 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 5724 . . 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 2493 . 2  |-  ( C  e.  _V  -> tpos  H  =  (  Hom  `  O
) )
28 tpos0 6501 . . 3  |- tpos  (/)  =  (/)
29 fvprc 5714 . . . . 5  |-  ( -.  C  e.  _V  ->  (  Hom  `  C )  =  (/) )
3016, 29syl5eq 2479 . . . 4  |-  ( -.  C  e.  _V  ->  H  =  (/) )
3130tposeqd 6474 . . 3  |-  ( -.  C  e.  _V  -> tpos  H  = tpos  (/) )
32 fvprc 5714 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
3324, 32syl5eq 2479 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
3433fveq2d 5724 . . . 4  |-  ( -.  C  e.  _V  ->  (  Hom  `  O )  =  (  Hom  `  (/) ) )
35 df-hom 13545 . . . . 5  |-  Hom  = Slot ; 1 4
3635str0 13497 . . . 4  |-  (/)  =  (  Hom  `  (/) )
3734, 36syl6eqr 2485 . . 3  |-  ( -.  C  e.  _V  ->  (  Hom  `  O )  =  (/) )
3828, 31, 373eqtr4a 2493 . 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 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948   (/)c0 3620   <.cop 3809    X. cxp 4868   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340  tpos ctpos 6470   1c1 8983   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:  oppchom  13933
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-dec 10375  df-ndx 13464  df-slot 13465  df-sets 13467  df-hom 13545  df-cco 13546  df-oppc 13930
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