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Theorem xpcbas 15110
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 2454 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
5 eqid 2454 . . . . 5  |-  ( Hom  `  D )  =  ( Hom  `  D )
6 eqid 2454 . . . . 5  |-  (comp `  C )  =  (comp `  C )
7 eqid 2454 . . . . 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 2455 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  ( X  X.  Y ) )
11 eqidd 2455 . . . . 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 2455 . . . . 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 15109 . . . 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 14989 . . . 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 14341 . . . 4  |-  Base  = Slot  ( Base `  ndx )
16 snsstp1 4135 . . . 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 5812 . . . . . . 7  |-  ( Base `  C )  e.  _V
182, 17eqeltri 2538 . . . . . 6  |-  X  e. 
_V
19 fvex 5812 . . . . . . 7  |-  ( Base `  D )  e.  _V
203, 19eqeltri 2538 . . . . . 6  |-  Y  e. 
_V
2118, 20xpex 6621 . . . . 5  |-  ( X  X.  Y )  e. 
_V
2221a1i 11 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  e.  _V )
23 eqid 2454 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
2413, 14, 15, 16, 22, 23strfv3 14330 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( X  X.  Y ) )
2524eqcomd 2462 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  ( Base `  T ) )
26 base0 14334 . . 3  |-  (/)  =  (
Base `  (/) )
27 fvprc 5796 . . . . . 6  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
282, 27syl5eq 2507 . . . . 5  |-  ( -.  C  e.  _V  ->  X  =  (/) )
29 fvprc 5796 . . . . . 6  |-  ( -.  D  e.  _V  ->  (
Base `  D )  =  (/) )
303, 29syl5eq 2507 . . . . 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 5369 . . . 4  |-  ( ( X  X.  Y )  =  (/)  <->  ( X  =  (/)  \/  Y  =  (/) ) )
3431, 32, 333imtr4i 266 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( X  X.  Y
)  =  (/) )
35 fnxpc 15108 . . . . . . 7  |-  X.c  Fn  ( _V  X.  _V )
36 fndm 5621 . . . . . . 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 6360 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( C  X.c  D )  =  (/) )
391, 38syl5eq 2507 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  T  =  (/) )
4039fveq2d 5806 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( Base `  (/) ) )
4126, 34, 403eqtr4a 2521 . 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 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3748   {ctp 3992   <.cop 3994    X. cxp 4949   dom cdm 4951    Fn wfn 5524   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   1c1 9397   5c5 10488  ;cdc 10869   ndxcnx 14292   Basecbs 14295   Hom chom 14371  compcco 14372    X.c cxpc 15100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-hom 14384  df-cco 14385  df-xpc 15104
This theorem is referenced by:  xpchomfval  15111  xpccofval  15114  xpchom2  15118  xpcco2  15119  xpccatid  15120  1stfval  15123  2ndfval  15126  1stfcl  15129  2ndfcl  15130  prfcl  15135  prf1st  15136  prf2nd  15137  1st2ndprf  15138  catcxpccl  15139  xpcpropd  15140  evlfcl  15154  curf1cl  15160  curf2cl  15163  curfcl  15164  uncf1  15168  uncf2  15169  uncfcurf  15171  diag11  15175  diag12  15176  diag2  15177  curf2ndf  15179  hofcl  15191  yonedalem21  15205  yonedalem22  15210  yonedalem3b  15211  yonedalem3  15212  yonedainv  15213  yonffthlem  15214
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