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Theorem xpchomfval 15323
Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpchomfval.t  |-  T  =  ( C  X.c  D )
xpchomfval.y  |-  B  =  ( Base `  T
)
xpchomfval.h  |-  H  =  ( Hom  `  C
)
xpchomfval.j  |-  J  =  ( Hom  `  D
)
xpchomfval.k  |-  K  =  ( Hom  `  T
)
Assertion
Ref Expression
xpchomfval  |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
Distinct variable groups:    v, u, B    u, C, v    u, D, v    u, H, v   
u, J, v
Allowed substitution hints:    T( v, u)    K( v, u)

Proof of Theorem xpchomfval
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpchomfval.t . . . 4  |-  T  =  ( C  X.c  D )
2 eqid 2467 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2467 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 xpchomfval.h . . . 4  |-  H  =  ( Hom  `  C
)
5 xpchomfval.j . . . 4  |-  J  =  ( Hom  `  D
)
6 eqid 2467 . . . 4  |-  (comp `  C )  =  (comp `  C )
7 eqid 2467 . . . 4  |-  (comp `  D )  =  (comp `  D )
8 simpl 457 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  C  e.  _V )
9 simpr 461 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  D  e.  _V )
10 xpchomfval.y . . . . . 6  |-  B  =  ( Base `  T
)
111, 2, 3xpcbas 15322 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
1210, 11eqtr4i 2499 . . . . 5  |-  B  =  ( ( Base `  C
)  X.  ( Base `  D ) )
1312a1i 11 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  B  =  ( (
Base `  C )  X.  ( Base `  D
) ) )
14 eqidd 2468 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
15 eqidd 2468 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) ) y ) ,  f  e.  ( ( u  e.  B , 
v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) ) y ) ,  f  e.  ( ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15xpcval 15321 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  T  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
>. ,  <. (comp `  ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) y ) ,  f  e.  ( ( u  e.  B , 
v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
17 catstr 15201 . . 3  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
>. ,  <. (comp `  ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) y ) ,  f  e.  ( ( u  e.  B , 
v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } Struct  <. 1 , ; 1 5 >.
18 homid 14688 . . 3  |-  Hom  = Slot  ( Hom  `  ndx )
19 snsstp2 4185 . . 3  |-  { <. ( Hom  `  ndx ) ,  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
>. }  C_  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
>. ,  <. (comp `  ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) y ) ,  f  e.  ( ( u  e.  B , 
v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }
20 fvex 5882 . . . . . 6  |-  ( Base `  T )  e.  _V
2110, 20eqeltri 2551 . . . . 5  |-  B  e. 
_V
2221, 21mpt2ex 6872 . . . 4  |-  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) )  e.  _V
2322a1i 11 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )  e.  _V )
24 xpchomfval.k . . 3  |-  K  =  ( Hom  `  T
)
2516, 17, 18, 19, 23, 24strfv3 14542 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
26 mpt20 6362 . . . 4  |-  ( u  e.  (/) ,  v  e.  (/)  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) )  =  (/)
2726eqcomi 2480 . . 3  |-  (/)  =  ( u  e.  (/) ,  v  e.  (/)  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) )
28 fnxpc 15320 . . . . . . . 8  |-  X.c  Fn  ( _V  X.  _V )
29 fndm 5686 . . . . . . . 8  |-  (  X.c  Fn  ( _V  X.  _V )  ->  dom  X.c  =  ( _V  X.  _V ) )
3028, 29ax-mp 5 . . . . . . 7  |-  dom  X.c  =  ( _V  X.  _V )
3130ndmov 6454 . . . . . 6  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( C  X.c  D )  =  (/) )
321, 31syl5eq 2520 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  T  =  (/) )
3332fveq2d 5876 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Hom  `  T
)  =  ( Hom  `  (/) ) )
3418str0 14545 . . . 4  |-  (/)  =  ( Hom  `  (/) )
3533, 24, 343eqtr4g 2533 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  K  =  (/) )
3632fveq2d 5876 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( Base `  (/) ) )
37 base0 14546 . . . . 5  |-  (/)  =  (
Base `  (/) )
3836, 10, 373eqtr4g 2533 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  B  =  (/) )
39 eqidd 2468 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) )  =  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
4038, 38, 39mpt2eq123dv 6354 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )  =  ( u  e.  (/) ,  v  e.  (/)  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) ) )
4127, 35, 403eqtr4a 2534 . 2  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
4225, 41pm2.61i 164 1  |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   {ctp 4037   <.cop 4039    X. cxp 5003   dom cdm 5005    Fn wfn 5589   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   1c1 9505   5c5 10600  ;cdc 10988   ndxcnx 14504   Basecbs 14507   Hom chom 14583  compcco 14584    X.c cxpc 15312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-hom 14596  df-cco 14597  df-xpc 15316
This theorem is referenced by:  xpchom  15324  relxpchom  15325  xpccofval  15326  catcxpccl  15351  xpcpropd  15352
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