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Theorem xpcbas 16006
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 2428 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
5 eqid 2428 . . . . 5  |-  ( Hom  `  D )  =  ( Hom  `  D )
6 eqid 2428 . . . . 5  |-  (comp `  C )  =  (comp `  C )
7 eqid 2428 . . . . 5  |-  (comp `  D )  =  (comp `  D )
8 simpl 458 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  C  e.  _V )
9 simpr 462 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  D  e.  _V )
10 eqidd 2429 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  ( X  X.  Y ) )
11 eqidd 2429 . . . . 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 2429 . . . . 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 16005 . . . 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 15805 . . . 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 15112 . . . 4  |-  Base  = Slot  ( Base `  ndx )
16 snsstp1 4094 . . . 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 5835 . . . . . . 7  |-  ( Base `  C )  e.  _V
182, 17eqeltri 2502 . . . . . 6  |-  X  e. 
_V
19 fvex 5835 . . . . . . 7  |-  ( Base `  D )  e.  _V
203, 19eqeltri 2502 . . . . . 6  |-  Y  e. 
_V
2118, 20xpex 6553 . . . . 5  |-  ( X  X.  Y )  e. 
_V
2221a1i 11 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  e.  _V )
23 eqid 2428 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
2413, 14, 15, 16, 22, 23strfv3 15101 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( X  X.  Y ) )
2524eqcomd 2434 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  ( Base `  T ) )
26 base0 15105 . . 3  |-  (/)  =  (
Base `  (/) )
27 fvprc 5819 . . . . . 6  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
282, 27syl5eq 2474 . . . . 5  |-  ( -.  C  e.  _V  ->  X  =  (/) )
29 fvprc 5819 . . . . . 6  |-  ( -.  D  e.  _V  ->  (
Base `  D )  =  (/) )
303, 29syl5eq 2474 . . . . 5  |-  ( -.  D  e.  _V  ->  Y  =  (/) )
3128, 30orim12i 518 . . . 4  |-  ( ( -.  C  e.  _V  \/  -.  D  e.  _V )  ->  ( X  =  (/)  \/  Y  =  (/) ) )
32 ianor 490 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  <->  ( -.  C  e.  _V  \/  -.  D  e.  _V ) )
33 xpeq0 5219 . . . 4  |-  ( ( X  X.  Y )  =  (/)  <->  ( X  =  (/)  \/  Y  =  (/) ) )
3431, 32, 333imtr4i 269 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  (/) )
35 fnxpc 16004 . . . . . . 7  |-  X.c  Fn  ( _V  X.  _V )
36 fndm 5636 . . . . . . 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 6411 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( C  X.c  D )  =  (/) )
391, 38syl5eq 2474 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  T  =  (/) )
4039fveq2d 5829 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( Base `  (/) ) )
4126, 34, 403eqtr4a 2488 . 2  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  ( Base `  T ) )
4225, 41pm2.61i 167 1  |-  ( X  X.  Y )  =  ( Base `  T
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3022   (/)c0 3704   {ctp 3945   <.cop 3947    X. cxp 4794   dom cdm 4796    Fn wfn 5539   ` cfv 5544  (class class class)co 6249    |-> cmpt2 6251   1stc1st 6749   2ndc2nd 6750   1c1 9491   5c5 10613  ;cdc 11002   ndxcnx 15061   Basecbs 15064   Hom chom 15144  compcco 15145    X.c cxpc 15996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-10 10627  df-n0 10821  df-z 10889  df-dec 11003  df-uz 11111  df-fz 11736  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-hom 15157  df-cco 15158  df-xpc 16000
This theorem is referenced by:  xpchomfval  16007  xpccofval  16010  xpchom2  16014  xpcco2  16015  xpccatid  16016  1stfval  16019  2ndfval  16022  1stfcl  16025  2ndfcl  16026  prfcl  16031  prf1st  16032  prf2nd  16033  1st2ndprf  16034  catcxpccl  16035  xpcpropd  16036  evlfcl  16050  curf1cl  16056  curf2cl  16059  curfcl  16060  uncf1  16064  uncf2  16065  uncfcurf  16067  diag11  16071  diag12  16072  diag2  16073  curf2ndf  16075  hofcl  16087  yonedalem21  16101  yonedalem22  16106  yonedalem3b  16107  yonedalem3  16108  yonedainv  16109  yonffthlem  16110
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