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Theorem s1co 12461
Description: Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1co  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  <" ( F `  S ) "> )

Proof of Theorem s1co
StepHypRef Expression
1 s1val 12290 . . . . 5  |-  ( S  e.  A  ->  <" S ">  =  { <. 0 ,  S >. } )
2 0cn 9378 . . . . . 6  |-  0  e.  CC
3 xpsng 5884 . . . . . 6  |-  ( ( 0  e.  CC  /\  S  e.  A )  ->  ( { 0 }  X.  { S }
)  =  { <. 0 ,  S >. } )
42, 3mpan 670 . . . . 5  |-  ( S  e.  A  ->  ( { 0 }  X.  { S } )  =  { <. 0 ,  S >. } )
51, 4eqtr4d 2478 . . . 4  |-  ( S  e.  A  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
65adantr 465 . . 3  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
76coeq2d 5002 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  ( F  o.  ( { 0 }  X.  { S }
) ) )
8 ffn 5559 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
9 id 22 . . . 4  |-  ( S  e.  A  ->  S  e.  A )
10 fcoconst 5880 . . . 4  |-  ( ( F  Fn  A  /\  S  e.  A )  ->  ( F  o.  ( { 0 }  X.  { S } ) )  =  ( { 0 }  X.  { ( F `  S ) } ) )
118, 9, 10syl2anr 478 . . 3  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  ( { 0 }  X.  { S } ) )  =  ( { 0 }  X.  { ( F `  S ) } ) )
12 fvex 5701 . . . . 5  |-  ( F `
 S )  e. 
_V
13 s1val 12290 . . . . 5  |-  ( ( F `  S )  e.  _V  ->  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. } )
1412, 13ax-mp 5 . . . 4  |-  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. }
15 c0ex 9380 . . . . 5  |-  0  e.  _V
1615, 12xpsn 5885 . . . 4  |-  ( { 0 }  X.  {
( F `  S
) } )  =  { <. 0 ,  ( F `  S )
>. }
1714, 16eqtr4i 2466 . . 3  |-  <" ( F `  S ) ">  =  ( { 0 }  X.  {
( F `  S
) } )
1811, 17syl6reqr 2494 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" ( F `
 S ) ">  =  ( F  o.  ( { 0 }  X.  { S } ) ) )
197, 18eqtr4d 2478 1  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  <" ( F `  S ) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972   {csn 3877   <.cop 3883    X. cxp 4838    o. ccom 4844    Fn wfn 5413   -->wf 5414   ` cfv 5418   CCcc 9280   0cc0 9282   <"cs1 12224
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 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-mulcl 9344  ax-i2m1 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-s1 12232
This theorem is referenced by:  cats1co  12483  s2co  12530  frmdgsum  15540  frmdup2  15543  efginvrel2  16224  vrgpinv  16266  frgpup2  16273
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