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Theorem xpchomfval 14278
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 2438 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2438 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 xpchomfval.h . . . 4  |-  H  =  (  Hom  `  C
)
5 xpchomfval.j . . . 4  |-  J  =  (  Hom  `  D
)
6 eqid 2438 . . . 4  |-  (comp `  C )  =  (comp `  C )
7 eqid 2438 . . . 4  |-  (comp `  D )  =  (comp `  D )
8 simpl 445 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  C  e.  _V )
9 simpr 449 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  D  e.  _V )
10 xpchomfval.y . . . . . 6  |-  B  =  ( Base `  T
)
111, 2, 3xpcbas 14277 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
1210, 11eqtr4i 2461 . . . . 5  |-  B  =  ( ( Base `  C
)  X.  ( Base `  D ) )
1312a1i 11 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  B  =  ( (
Base `  C )  X.  ( Base `  D
) ) )
14 eqidd 2439 . . . 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 2439 . . . 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 14276 . . 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 14156 . . 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 13645 . . 3  |-  Hom  = Slot  (  Hom  `  ndx )
19 snsstp2 3952 . . 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 5744 . . . . . 6  |-  ( Base `  T )  e.  _V
2110, 20eqeltri 2508 . . . . 5  |-  B  e. 
_V
2221, 21mpt2ex 6427 . . . 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 13504 . 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 6429 . . . 4  |-  ( u  e.  (/) ,  v  e.  (/)  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) )  =  (/)
2726eqcomi 2442 . . 3  |-  (/)  =  ( u  e.  (/) ,  v  e.  (/)  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) )
28 fnxpc 14275 . . . . . . . 8  |-  X.c  Fn  ( _V  X.  _V )
29 fndm 5546 . . . . . . . 8  |-  (  X.c  Fn  ( _V  X.  _V )  ->  dom  X.c  =  ( _V  X.  _V ) )
3028, 29ax-mp 8 . . . . . . 7  |-  dom  X.c  =  ( _V  X.  _V )
3130ndmov 6233 . . . . . 6  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( C  X.c  D )  =  (/) )
321, 31syl5eq 2482 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  T  =  (/) )
3332fveq2d 5734 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  (  Hom  `  T
)  =  (  Hom  `  (/) ) )
3418str0 13507 . . . 4  |-  (/)  =  (  Hom  `  (/) )
3533, 24, 343eqtr4g 2495 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  K  =  (/) )
3632fveq2d 5734 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( Base `  (/) ) )
37 base0 13508 . . . . 5  |-  (/)  =  (
Base `  (/) )
3836, 10, 373eqtr4g 2495 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  B  =  (/) )
39 eqidd 2439 . . . 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 6138 . . 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 2496 . 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 159 1  |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   {ctp 3818   <.cop 3819    X. cxp 4878   dom cdm 4880    Fn wfn 5451   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350   1c1 8993   5c5 10054  ;cdc 10384   ndxcnx 13468   Basecbs 13471    Hom chom 13542  compcco 13543    X.c cxpc 14267
This theorem is referenced by:  xpchom  14279  relxpchom  14280  xpccofval  14281  catcxpccl  14306  xpcpropd  14307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-hom 13555  df-cco 13556  df-xpc 14271
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