Theorem 1strbas 13165

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

Step	Hyp	Ref	Expression
1		baseslid 13105	. 2 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
2		1str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. }
3		basendxnn 13103	. . . . 5 |- ( Base ` ndx ) e. NN
4		opexg 4314	. . . . 5 |- ( ( ( Base ` ndx ) e. NN /\ B e. V ) -> <. ( Base ` ndx ) , B >. e. _V )
5	3, 4	mpan 424	. . . 4 |- ( B e. V -> <. ( Base ` ndx ) , B >. e. _V )
6		snexg 4268	. . . 4 |- ( <. ( Base ` ndx ) , B >. e. _V -> { <. ( Base ` ndx ) , B >. } e. _V )
7	5, 6	syl 14	. . 3 |- ( B e. V -> { <. ( Base ` ndx ) , B >. } e. _V )
8	2, 7	eqeltrid 2316	. 2 |- ( B e. V -> G e. _V )
9		funsng 5367	. . . 4 |- ( ( ( Base ` ndx ) e. NN /\ B e. V ) -> Fun { <. ( Base ` ndx ) , B >. } )
10	3, 9	mpan 424	. . 3 |- ( B e. V -> Fun { <. ( Base ` ndx ) , B >. } )
11	2	funeqi 5339	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. } )
12	10, 11	sylibr 134	. 2 |- ( B e. V -> Fun G )
13		snidg 3695	. . . 4 |- ( <. ( Base ` ndx ) , B >. e. _V -> <. ( Base ` ndx ) , B >. e. { <. ( Base ` ndx ) , B >. } )
14	5, 13	syl 14	. . 3 |- ( B e. V -> <. ( Base ` ndx ) , B >. e. { <. ( Base ` ndx ) , B >. } )
15	14, 2	eleqtrrdi 2323	. 2 |- ( B e. V -> <. ( Base ` ndx ) , B >. e. G )
16	1, 8, 12, 15	strslfvd 13089	1 |- ( B e. V -> B = ( Base ` G ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   <.cop 3669   Fun wfun 5312   ` cfv 5318   NNcn 9121   ndxcnx 13044   Basecbs 13047

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-inn 9122  df-ndx 13050  df-slot 13051  df-base 13053

This theorem is referenced by: (None)