Theorem 1strbas 12735

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 12675	. 2 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
2		1str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. }
3		basendxnn 12674	. . . . 5 |- ( Base ` ndx ) e. NN
4		opexg 4257	. . . . 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 4213	. . . 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 2280	. 2 |- ( B e. V -> G e. _V )
9		funsng 5300	. . . 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 5275	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. } )
12	10, 11	sylibr 134	. 2 |- ( B e. V -> Fun G )
13		snidg 3647	. . . 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 2287	. 2 |- ( B e. V -> <. ( Base ` ndx ) , B >. e. G )
16	1, 8, 12, 15	strslfvd 12660	1 |- ( B e. V -> B = ( Base ` G ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   {csn 3618   <.cop 3621   Fun wfun 5248   ` cfv 5254   NNcn 8982   ndxcnx 12615   Basecbs 12618

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624

This theorem is referenced by: (None)