MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s1co Unicode version

Theorem s1co 11488
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 11438 . . . . 5  |-  ( S  e.  A  ->  <" S ">  =  { <. 0 ,  S >. } )
2 0cn 8831 . . . . . 6  |-  0  e.  CC
3 xpsng 5699 . . . . . 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 2318 . . . 4  |-  ( S  e.  A  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
65adantr 451 . . 3  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" S ">  =  ( { 0 }  X.  { S } ) )
76coeq2d 4846 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  ( F  o.  ( { 0 }  X.  { S }
) ) )
8 ffn 5389 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
9 id 19 . . . 4  |-  ( S  e.  A  ->  S  e.  A )
10 fcoconst 5695 . . . 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 5539 . . . . 5  |-  ( F `
 S )  e. 
_V
13 s1val 11438 . . . . 5  |-  ( ( F `  S )  e.  _V  ->  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. } )
1412, 13ax-mp 8 . . . 4  |-  <" ( F `  S ) ">  =  { <. 0 ,  ( F `  S ) >. }
15 c0ex 8832 . . . . 5  |-  0  e.  _V
1615, 12xpsn 5700 . . . 4  |-  ( { 0 }  X.  {
( F `  S
) } )  =  { <. 0 ,  ( F `  S )
>. }
1714, 16eqtr4i 2306 . . 3  |-  <" ( F `  S ) ">  =  ( { 0 }  X.  {
( F `  S
) } )
1811, 17syl6reqr 2334 . 2  |-  ( ( S  e.  A  /\  F : A --> B )  ->  <" ( F `
 S ) ">  =  ( F  o.  ( { 0 }  X.  { S } ) ) )
197, 18eqtr4d 2318 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 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643    X. cxp 4687    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255   CCcc 8735   0cc0 8737   <"cs1 11405
This theorem is referenced by:  cats1co  11506  s2co  11547  frmdgsum  14484  frmdup2  14487  efginvrel2  15036  vrgpinv  15078  frgpup2  15085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-s1 11411
  Copyright terms: Public domain W3C validator