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Theorem s1co 11794
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 11744 . . . . 5  |-  ( S  e.  A  ->  <" S ">  =  { <. 0 ,  S >. } )
2 0cn 9076 . . . . . 6  |-  0  e.  CC
3 xpsng 5901 . . . . . 6  |-  ( ( 0  e.  CC  /\  S  e.  A )  ->  ( { 0 }  X.  { S }
)  =  { <. 0 ,  S >. } )
42, 3mpan 652 . . . . 5  |-  ( S  e.  A  ->  ( { 0 }  X.  { S } )  =  { <. 0 ,  S >. } )
51, 4eqtr4d 2470 . . . 4  |-  ( S  e.  A  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
65adantr 452 . . 3  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
76coeq2d 5027 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  ( F  o.  ( { 0 }  X.  { S }
) ) )
8 ffn 5583 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
9 id 20 . . . 4  |-  ( S  e.  A  ->  S  e.  A )
10 fcoconst 5897 . . . 4  |-  ( ( F  Fn  A  /\  S  e.  A )  ->  ( F  o.  ( { 0 }  X.  { S } ) )  =  ( { 0 }  X.  { ( F `  S ) } ) )
118, 9, 10syl2anr 465 . . 3  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  ( { 0 }  X.  { S } ) )  =  ( { 0 }  X.  { ( F `  S ) } ) )
12 fvex 5734 . . . . 5  |-  ( F `
 S )  e. 
_V
13 s1val 11744 . . . . 5  |-  ( ( F `  S )  e.  _V  ->  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. } )
1412, 13ax-mp 8 . . . 4  |-  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. }
15 c0ex 9077 . . . . 5  |-  0  e.  _V
1615, 12xpsn 5902 . . . 4  |-  ( { 0 }  X.  {
( F `  S
) } )  =  { <. 0 ,  ( F `  S )
>. }
1714, 16eqtr4i 2458 . . 3  |-  <" ( F `  S ) ">  =  ( { 0 }  X.  {
( F `  S
) } )
1811, 17syl6reqr 2486 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" ( F `
 S ) ">  =  ( F  o.  ( { 0 }  X.  { S } ) ) )
197, 18eqtr4d 2470 1  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  <" ( F `  S ) "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   <.cop 3809    X. cxp 4868    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446   CCcc 8980   0cc0 8982   <"cs1 11711
This theorem is referenced by:  cats1co  11812  s2co  11859  frmdgsum  14799  frmdup2  14802  efginvrel2  15351  vrgpinv  15393  frgpup2  15400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-mulcl 9044  ax-i2m1 9050
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-s1 11717
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