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Theorem s1co 11722
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 11672 . . . . 5  |-  ( S  e.  A  ->  <" S ">  =  { <. 0 ,  S >. } )
2 0cn 9010 . . . . . 6  |-  0  e.  CC
3 xpsng 5841 . . . . . 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 2415 . . . 4  |-  ( S  e.  A  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
65adantr 452 . . 3  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
76coeq2d 4968 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  ( F  o.  ( { 0 }  X.  { S }
) ) )
8 ffn 5524 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
9 id 20 . . . 4  |-  ( S  e.  A  ->  S  e.  A )
10 fcoconst 5837 . . . 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 5675 . . . . 5  |-  ( F `
 S )  e. 
_V
13 s1val 11672 . . . . 5  |-  ( ( F `  S )  e.  _V  ->  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. } )
1412, 13ax-mp 8 . . . 4  |-  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. }
15 c0ex 9011 . . . . 5  |-  0  e.  _V
1615, 12xpsn 5842 . . . 4  |-  ( { 0 }  X.  {
( F `  S
) } )  =  { <. 0 ,  ( F `  S )
>. }
1714, 16eqtr4i 2403 . . 3  |-  <" ( F `  S ) ">  =  ( { 0 }  X.  {
( F `  S
) } )
1811, 17syl6reqr 2431 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" ( F `
 S ) ">  =  ( F  o.  ( { 0 }  X.  { S } ) ) )
197, 18eqtr4d 2415 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 1649    e. wcel 1717   _Vcvv 2892   {csn 3750   <.cop 3753    X. cxp 4809    o. ccom 4815    Fn wfn 5382   -->wf 5383   ` cfv 5387   CCcc 8914   0cc0 8916   <"cs1 11639
This theorem is referenced by:  cats1co  11740  s2co  11787  frmdgsum  14727  frmdup2  14730  efginvrel2  15279  vrgpinv  15321  frgpup2  15328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-mulcl 8978  ax-i2m1 8984
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-s1 11645
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