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Theorem 1strbas 11985
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 11942 . 2  |-  ( Base 
= Slot  ( Base `  ndx )  /\  ( Base `  ndx )  e.  NN )
2 1str.g . . 3  |-  G  =  { <. ( Base `  ndx ) ,  B >. }
3 basendxnn 11941 . . . . 5  |-  ( Base `  ndx )  e.  NN
4 opexg 4120 . . . . 5  |-  ( ( ( Base `  ndx )  e.  NN  /\  B  e.  V )  ->  <. ( Base `  ndx ) ,  B >.  e.  _V )
53, 4mpan 420 . . . 4  |-  ( B  e.  V  ->  <. ( Base `  ndx ) ,  B >.  e.  _V )
6 snexg 4078 . . . 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 2204 . 2  |-  ( B  e.  V  ->  G  e.  _V )
9 funsng 5139 . . . 4  |-  ( ( ( Base `  ndx )  e.  NN  /\  B  e.  V )  ->  Fun  {
<. ( Base `  ndx ) ,  B >. } )
103, 9mpan 420 . . 3  |-  ( B  e.  V  ->  Fun  {
<. ( Base `  ndx ) ,  B >. } )
112funeqi 5114 . . 3  |-  ( Fun 
G  <->  Fun  { <. ( Base `  ndx ) ,  B >. } )
1210, 11sylibr 133 . 2  |-  ( B  e.  V  ->  Fun  G )
13 snidg 3524 . . . 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 2211 . 2  |-  ( B  e.  V  ->  <. ( Base `  ndx ) ,  B >.  e.  G
)
161, 8, 12, 15strslfvd 11927 1  |-  ( B  e.  V  ->  B  =  ( Base `  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   _Vcvv 2660   {csn 3497   <.cop 3500   Fun wfun 5087   ` cfv 5093   NNcn 8688   ndxcnx 11883   Basecbs 11886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fv 5101  df-inn 8689  df-ndx 11889  df-slot 11890  df-base 11892
This theorem is referenced by: (None)
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