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Theorem oppchomfval 13617
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 oppchom.h . . . . . . 7  |-  H  =  (  Hom  `  C
)
2 fvex 5539 . . . . . . 7  |-  (  Hom  `  C )  e.  _V
31, 2eqeltri 2353 . . . . . 6  |-  H  e. 
_V
43tposex 6268 . . . . 5  |- tpos  H  e. 
_V
5 df-hom 13232 . . . . . . 7  |-  Hom  = Slot ; 1 4
6 1nn0 9981 . . . . . . . 8  |-  1  e.  NN0
7 4nn 9879 . . . . . . . 8  |-  4  e.  NN
86, 7decnncl 10137 . . . . . . 7  |- ; 1 4  e.  NN
95, 8ndxid 13169 . . . . . 6  |-  Hom  = Slot  (  Hom  `  ndx )
109setsid 13187 . . . . 5  |-  ( ( C  e.  _V  /\ tpos  H  e.  _V )  -> tpos  H  =  (  Hom  `  ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. )
) )
114, 10mpan2 652 . . . 4  |-  ( C  e.  _V  -> tpos  H  =  (  Hom  `  ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. )
) )
128nnrei 9755 . . . . . . 7  |- ; 1 4  e.  RR
13 4nn0 9984 . . . . . . . 8  |-  4  e.  NN0
14 5nn 9880 . . . . . . . 8  |-  5  e.  NN
15 4lt5 9892 . . . . . . . 8  |-  4  <  5
166, 13, 14, 15declt 10145 . . . . . . 7  |- ; 1 4  < ; 1 5
1712, 16ltneii 8931 . . . . . 6  |- ; 1 4  =/= ; 1 5
185, 8ndxarg 13168 . . . . . . 7  |-  (  Hom  `  ndx )  = ; 1 4
19 df-cco 13233 . . . . . . . 8  |- comp  = Slot ; 1 5
206, 14decnncl 10137 . . . . . . . 8  |- ; 1 5  e.  NN
2119, 20ndxarg 13168 . . . . . . 7  |-  (comp `  ndx )  = ; 1 5
2218, 21neeq12i 2458 . . . . . 6  |-  ( (  Hom  `  ndx )  =/=  (comp `  ndx )  <-> ; 1 4  =/= ; 1 5 )
2317, 22mpbir 200 . . . . 5  |-  (  Hom  `  ndx )  =/=  (comp ` 
ndx )
249, 23setsnid 13188 . . . 4  |-  (  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 ) ) )
>. ) )
2511, 24syl6eq 2331 . . 3  |-  ( C  e.  _V  -> 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 ) ) )
>. ) ) )
26 eqid 2283 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
27 eqid 2283 . . . . 5  |-  (comp `  C )  =  (comp `  C )
28 oppchom.o . . . . 5  |-  O  =  (oppCat `  C )
2926, 1, 27, 28oppcval 13616 . . . 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 ) ) )
>. ) )
3029fveq2d 5529 . . 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 ) ) )
>. ) ) )
3125, 30eqtr4d 2318 . 2  |-  ( C  e.  _V  -> tpos  H  =  (  Hom  `  O
) )
32 tpos0 6264 . . 3  |- tpos  (/)  =  (/)
33 fvprc 5519 . . . . 5  |-  ( -.  C  e.  _V  ->  (  Hom  `  C )  =  (/) )
341, 33syl5eq 2327 . . . 4  |-  ( -.  C  e.  _V  ->  H  =  (/) )
35 tposeq 6236 . . . 4  |-  ( H  =  (/)  -> tpos  H  = tpos  (/) )
3634, 35syl 15 . . 3  |-  ( -.  C  e.  _V  -> tpos  H  = tpos  (/) )
37 fvprc 5519 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
3828, 37syl5eq 2327 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
3938fveq2d 5529 . . . 4  |-  ( -.  C  e.  _V  ->  (  Hom  `  O )  =  (  Hom  `  (/) ) )
405str0 13184 . . . 4  |-  (/)  =  (  Hom  `  (/) )
4139, 40syl6eqr 2333 . . 3  |-  ( -.  C  e.  _V  ->  (  Hom  `  O )  =  (/) )
4232, 36, 413eqtr4a 2341 . 2  |-  ( -.  C  e.  _V  -> tpos  H  =  (  Hom  `  O
) )
4331, 42pm2.61i 156 1  |- tpos  H  =  (  Hom  `  O
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   <.cop 3643    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121  tpos ctpos 6233   1c1 8738   4c4 9797   5c5 9798  ;cdc 10124   ndxcnx 13145   sSet csts 13146   Basecbs 13148    Hom chom 13219  compcco 13220  oppCatcoppc 13614
This theorem is referenced by:  oppchom  13618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-dec 10125  df-ndx 13151  df-slot 13152  df-sets 13154  df-hom 13232  df-cco 13233  df-oppc 13615
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