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Theorem xpcbas 14980
Description: Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
Hypotheses
Ref Expression
xpcbas.t  |-  T  =  ( C  X.c  D )
xpcbas.x  |-  X  =  ( Base `  C
)
xpcbas.y  |-  Y  =  ( Base `  D
)
Assertion
Ref Expression
xpcbas  |-  ( X  X.  Y )  =  ( Base `  T
)

Proof of Theorem xpcbas
Dummy variables  f 
g  u  v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcbas.t . . . . 5  |-  T  =  ( C  X.c  D )
2 xpcbas.x . . . . 5  |-  X  =  ( Base `  C
)
3 xpcbas.y . . . . 5  |-  Y  =  ( Base `  D
)
4 eqid 2438 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
5 eqid 2438 . . . . 5  |-  ( Hom  `  D )  =  ( Hom  `  D )
6 eqid 2438 . . . . 5  |-  (comp `  C )  =  (comp `  C )
7 eqid 2438 . . . . 5  |-  (comp `  D )  =  (comp `  D )
8 simpl 457 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  C  e.  _V )
9 simpr 461 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  D  e.  _V )
10 eqidd 2439 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  ( X  X.  Y ) )
11 eqidd 2439 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  ( ( ( 1st `  u ) ( Hom  `  C ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) )  =  ( u  e.  ( X  X.  Y
) ,  v  e.  ( X  X.  Y
)  |->  ( ( ( 1st `  u ) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) ) )
12 eqidd 2439 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( x  e.  ( ( X  X.  Y
)  X.  ( X  X.  Y ) ) ,  y  e.  ( X  X.  Y ) 
|->  ( g  e.  ( ( 2nd `  x
) ( u  e.  ( X  X.  Y
) ,  v  e.  ( X  X.  Y
)  |->  ( ( ( 1st `  u ) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) ) y ) ,  f  e.  ( ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  ( ( ( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 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.  ( ( X  X.  Y )  X.  ( X  X.  Y ) ) ,  y  e.  ( X  X.  Y )  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  ( ( ( 1st `  u ) ( Hom  `  C ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) ) y ) ,  f  e.  ( ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  ( ( ( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 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 ) ) >.
) ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12xpcval 14979 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  T  =  { <. (
Base `  ndx ) ,  ( X  X.  Y
) >. ,  <. ( Hom  `  ndx ) ,  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  ( ( ( 1st `  u ) ( Hom  `  C ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) )
>. ,  <. (comp `  ndx ) ,  ( x  e.  ( ( X  X.  Y )  X.  ( X  X.  Y
) ) ,  y  e.  ( X  X.  Y )  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  ( ( ( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) ) y ) ,  f  e.  ( ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  ( ( ( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 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 ) ) >.
) ) >. } )
14 catstr 14859 . . . 4  |-  { <. (
Base `  ndx ) ,  ( X  X.  Y
) >. ,  <. ( Hom  `  ndx ) ,  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  ( ( ( 1st `  u ) ( Hom  `  C ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) )
>. ,  <. (comp `  ndx ) ,  ( x  e.  ( ( X  X.  Y )  X.  ( X  X.  Y
) ) ,  y  e.  ( X  X.  Y )  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  ( ( ( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) ) y ) ,  f  e.  ( ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  ( ( ( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 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 >.
15 baseid 14212 . . . 4  |-  Base  = Slot  ( Base `  ndx )
16 snsstp1 4019 . . . 4  |-  { <. (
Base `  ndx ) ,  ( X  X.  Y
) >. }  C_  { <. (
Base `  ndx ) ,  ( X  X.  Y
) >. ,  <. ( Hom  `  ndx ) ,  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  ( ( ( 1st `  u ) ( Hom  `  C ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) )
>. ,  <. (comp `  ndx ) ,  ( x  e.  ( ( X  X.  Y )  X.  ( X  X.  Y
) ) ,  y  e.  ( X  X.  Y )  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  ( ( ( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) ) y ) ,  f  e.  ( ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  ( ( ( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 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 fvex 5696 . . . . . . 7  |-  ( Base `  C )  e.  _V
182, 17eqeltri 2508 . . . . . 6  |-  X  e. 
_V
19 fvex 5696 . . . . . . 7  |-  ( Base `  D )  e.  _V
203, 19eqeltri 2508 . . . . . 6  |-  Y  e. 
_V
2118, 20xpex 6503 . . . . 5  |-  ( X  X.  Y )  e. 
_V
2221a1i 11 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  e.  _V )
23 eqid 2438 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
2413, 14, 15, 16, 22, 23strfv3 14201 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( X  X.  Y ) )
2524eqcomd 2443 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  ( Base `  T ) )
26 base0 14205 . . 3  |-  (/)  =  (
Base `  (/) )
27 fvprc 5680 . . . . . 6  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
282, 27syl5eq 2482 . . . . 5  |-  ( -.  C  e.  _V  ->  X  =  (/) )
29 fvprc 5680 . . . . . 6  |-  ( -.  D  e.  _V  ->  (
Base `  D )  =  (/) )
303, 29syl5eq 2482 . . . . 5  |-  ( -.  D  e.  _V  ->  Y  =  (/) )
3128, 30orim12i 516 . . . 4  |-  ( ( -.  C  e.  _V  \/  -.  D  e.  _V )  ->  ( X  =  (/)  \/  Y  =  (/) ) )
32 ianor 488 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  <->  ( -.  C  e.  _V  \/  -.  D  e.  _V ) )
33 xpeq0 5253 . . . 4  |-  ( ( X  X.  Y )  =  (/)  <->  ( X  =  (/)  \/  Y  =  (/) ) )
3431, 32, 333imtr4i 266 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  (/) )
35 fnxpc 14978 . . . . . . 7  |-  X.c  Fn  ( _V  X.  _V )
36 fndm 5505 . . . . . . 7  |-  (  X.c  Fn  ( _V  X.  _V )  ->  dom  X.c  =  ( _V  X.  _V ) )
3735, 36ax-mp 5 . . . . . 6  |-  dom  X.c  =  ( _V  X.  _V )
3837ndmov 6242 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( C  X.c  D )  =  (/) )
391, 38syl5eq 2482 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  T  =  (/) )
4039fveq2d 5690 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( Base `  (/) ) )
4126, 34, 403eqtr4a 2496 . 2  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  ( Base `  T ) )
4225, 41pm2.61i 164 1  |-  ( X  X.  Y )  =  ( Base `  T
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   (/)c0 3632   {ctp 3876   <.cop 3878    X. cxp 4833   dom cdm 4835    Fn wfn 5408   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571   1c1 9275   5c5 10366  ;cdc 10747   ndxcnx 14163   Basecbs 14166   Hom chom 14241  compcco 14242    X.c cxpc 14970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-hom 14254  df-cco 14255  df-xpc 14974
This theorem is referenced by:  xpchomfval  14981  xpccofval  14984  xpchom2  14988  xpcco2  14989  xpccatid  14990  1stfval  14993  2ndfval  14996  1stfcl  14999  2ndfcl  15000  prfcl  15005  prf1st  15006  prf2nd  15007  1st2ndprf  15008  catcxpccl  15009  xpcpropd  15010  evlfcl  15024  curf1cl  15030  curf2cl  15033  curfcl  15034  uncf1  15038  uncf2  15039  uncfcurf  15041  diag11  15045  diag12  15046  diag2  15047  curf2ndf  15049  hofcl  15061  yonedalem21  15075  yonedalem22  15080  yonedalem3b  15081  yonedalem3  15082  yonedainv  15083  yonffthlem  15084
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