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Theorem s1co 11504
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 11454 . . . . 5  |-  ( S  e.  A  ->  <" S ">  =  { <. 0 ,  S >. } )
2 0cn 8847 . . . . . 6  |-  0  e.  CC
3 xpsng 5715 . . . . . 6  |-  ( ( 0  e.  CC  /\  S  e.  A )  ->  ( { 0 }  X.  { S }
)  =  { <. 0 ,  S >. } )
42, 3mpan 651 . . . . 5  |-  ( S  e.  A  ->  ( { 0 }  X.  { S } )  =  { <. 0 ,  S >. } )
51, 4eqtr4d 2331 . . . 4  |-  ( S  e.  A  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
65adantr 451 . . 3  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
76coeq2d 4862 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  ( F  o.  ( { 0 }  X.  { S }
) ) )
8 ffn 5405 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
9 id 19 . . . 4  |-  ( S  e.  A  ->  S  e.  A )
10 fcoconst 5711 . . . 4  |-  ( ( F  Fn  A  /\  S  e.  A )  ->  ( F  o.  ( { 0 }  X.  { S } ) )  =  ( { 0 }  X.  { ( F `  S ) } ) )
118, 9, 10syl2anr 464 . . 3  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  ( { 0 }  X.  { S } ) )  =  ( { 0 }  X.  { ( F `  S ) } ) )
12 fvex 5555 . . . . 5  |-  ( F `
 S )  e. 
_V
13 s1val 11454 . . . . 5  |-  ( ( F `  S )  e.  _V  ->  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. } )
1412, 13ax-mp 8 . . . 4  |-  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. }
15 c0ex 8848 . . . . 5  |-  0  e.  _V
1615, 12xpsn 5716 . . . 4  |-  ( { 0 }  X.  {
( F `  S
) } )  =  { <. 0 ,  ( F `  S )
>. }
1714, 16eqtr4i 2319 . . 3  |-  <" ( F `  S ) ">  =  ( { 0 }  X.  {
( F `  S
) } )
1811, 17syl6reqr 2347 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" ( F `
 S ) ">  =  ( F  o.  ( { 0 }  X.  { S } ) ) )
197, 18eqtr4d 2331 1  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  <" ( F `  S ) "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656    X. cxp 4703    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271   CCcc 8751   0cc0 8753   <"cs1 11421
This theorem is referenced by:  cats1co  11522  s2co  11563  frmdgsum  14500  frmdup2  14503  efginvrel2  15052  vrgpinv  15094  frgpup2  15101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-s1 11427
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