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Theorem 1strbas 12575
Description: The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
Hypothesis
Ref Expression
1str.g  |-  G  =  { <. ( Base `  ndx ) ,  B >. }
Assertion
Ref Expression
1strbas  |-  ( B  e.  V  ->  B  =  ( Base `  G
) )

Proof of Theorem 1strbas
StepHypRef Expression
1 baseslid 12518 . 2  |-  ( Base 
= Slot  ( Base `  ndx )  /\  ( Base `  ndx )  e.  NN )
2 1str.g . . 3  |-  G  =  { <. ( Base `  ndx ) ,  B >. }
3 basendxnn 12517 . . . . 5  |-  ( Base `  ndx )  e.  NN
4 opexg 4228 . . . . 5  |-  ( ( ( Base `  ndx )  e.  NN  /\  B  e.  V )  ->  <. ( Base `  ndx ) ,  B >.  e.  _V )
53, 4mpan 424 . . . 4  |-  ( B  e.  V  ->  <. ( Base `  ndx ) ,  B >.  e.  _V )
6 snexg 4184 . . . 4  |-  ( <.
( Base `  ndx ) ,  B >.  e.  _V  ->  { <. ( Base `  ndx ) ,  B >. }  e.  _V )
75, 6syl 14 . . 3  |-  ( B  e.  V  ->  { <. (
Base `  ndx ) ,  B >. }  e.  _V )
82, 7eqeltrid 2264 . 2  |-  ( B  e.  V  ->  G  e.  _V )
9 funsng 5262 . . . 4  |-  ( ( ( Base `  ndx )  e.  NN  /\  B  e.  V )  ->  Fun  {
<. ( Base `  ndx ) ,  B >. } )
103, 9mpan 424 . . 3  |-  ( B  e.  V  ->  Fun  {
<. ( Base `  ndx ) ,  B >. } )
112funeqi 5237 . . 3  |-  ( Fun 
G  <->  Fun  { <. ( Base `  ndx ) ,  B >. } )
1210, 11sylibr 134 . 2  |-  ( B  e.  V  ->  Fun  G )
13 snidg 3621 . . . 4  |-  ( <.
( Base `  ndx ) ,  B >.  e.  _V  -> 
<. ( Base `  ndx ) ,  B >.  e. 
{ <. ( Base `  ndx ) ,  B >. } )
145, 13syl 14 . . 3  |-  ( B  e.  V  ->  <. ( Base `  ndx ) ,  B >.  e.  { <. (
Base `  ndx ) ,  B >. } )
1514, 2eleqtrrdi 2271 . 2  |-  ( B  e.  V  ->  <. ( Base `  ndx ) ,  B >.  e.  G
)
161, 8, 12, 15strslfvd 12503 1  |-  ( B  e.  V  ->  B  =  ( Base `  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2737   {csn 3592   <.cop 3595   Fun wfun 5210   ` cfv 5216   NNcn 8918   ndxcnx 12458   Basecbs 12461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fv 5224  df-inn 8919  df-ndx 12464  df-slot 12465  df-base 12467
This theorem is referenced by: (None)
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