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Theorem oppchomfval 13633
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 5555 . . . . . . 7  |-  (  Hom  `  C )  e.  _V
31, 2eqeltri 2366 . . . . . 6  |-  H  e. 
_V
43tposex 6284 . . . . 5  |- tpos  H  e. 
_V
5 df-hom 13248 . . . . . . 7  |-  Hom  = Slot ; 1 4
6 1nn0 9997 . . . . . . . 8  |-  1  e.  NN0
7 4nn 9895 . . . . . . . 8  |-  4  e.  NN
86, 7decnncl 10153 . . . . . . 7  |- ; 1 4  e.  NN
95, 8ndxid 13185 . . . . . 6  |-  Hom  = Slot  (  Hom  `  ndx )
109setsid 13203 . . . . 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 9771 . . . . . . 7  |- ; 1 4  e.  RR
13 4nn0 10000 . . . . . . . 8  |-  4  e.  NN0
14 5nn 9896 . . . . . . . 8  |-  5  e.  NN
15 4lt5 9908 . . . . . . . 8  |-  4  <  5
166, 13, 14, 15declt 10161 . . . . . . 7  |- ; 1 4  < ; 1 5
1712, 16ltneii 8947 . . . . . 6  |- ; 1 4  =/= ; 1 5
185, 8ndxarg 13184 . . . . . . 7  |-  (  Hom  `  ndx )  = ; 1 4
19 df-cco 13249 . . . . . . . 8  |- comp  = Slot ; 1 5
206, 14decnncl 10153 . . . . . . . 8  |- ; 1 5  e.  NN
2119, 20ndxarg 13184 . . . . . . 7  |-  (comp `  ndx )  = ; 1 5
2218, 21neeq12i 2471 . . . . . 6  |-  ( (  Hom  `  ndx )  =/=  (comp `  ndx )  <-> ; 1 4  =/= ; 1 5 )
2317, 22mpbir 200 . . . . 5  |-  (  Hom  `  ndx )  =/=  (comp ` 
ndx )
249, 23setsnid 13204 . . . 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 2344 . . 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 2296 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
27 eqid 2296 . . . . 5  |-  (comp `  C )  =  (comp `  C )
28 oppchom.o . . . . 5  |-  O  =  (oppCat `  C )
2926, 1, 27, 28oppcval 13632 . . . 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 5545 . . 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 2331 . 2  |-  ( C  e.  _V  -> tpos  H  =  (  Hom  `  O
) )
32 tpos0 6280 . . 3  |- tpos  (/)  =  (/)
33 fvprc 5535 . . . . 5  |-  ( -.  C  e.  _V  ->  (  Hom  `  C )  =  (/) )
341, 33syl5eq 2340 . . . 4  |-  ( -.  C  e.  _V  ->  H  =  (/) )
35 tposeq 6252 . . . 4  |-  ( H  =  (/)  -> tpos  H  = tpos  (/) )
3634, 35syl 15 . . 3  |-  ( -.  C  e.  _V  -> tpos  H  = tpos  (/) )
37 fvprc 5535 . . . . . 6  |-  ( -.  C  e.  _V  ->  (oppCat `  C )  =  (/) )
3828, 37syl5eq 2340 . . . . 5  |-  ( -.  C  e.  _V  ->  O  =  (/) )
3938fveq2d 5545 . . . 4  |-  ( -.  C  e.  _V  ->  (  Hom  `  O )  =  (  Hom  `  (/) ) )
405str0 13200 . . . 4  |-  (/)  =  (  Hom  `  (/) )
4139, 40syl6eqr 2346 . . 3  |-  ( -.  C  e.  _V  ->  (  Hom  `  O )  =  (/) )
4232, 36, 413eqtr4a 2354 . 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 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   <.cop 3656    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137  tpos ctpos 6249   1c1 8754   4c4 9813   5c5 9814  ;cdc 10140   ndxcnx 13161   sSet csts 13162   Basecbs 13164    Hom chom 13235  compcco 13236  oppCatcoppc 13630
This theorem is referenced by:  oppchom  13634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-dec 10141  df-ndx 13167  df-slot 13168  df-sets 13170  df-hom 13248  df-cco 13249  df-oppc 13631
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