Theorem 1strbas 13199

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 13139	. 2 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
2		1str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. }
3		basendxnn 13137	. . . . 5 |- ( Base ` ndx ) e. NN
4		opexg 4320	. . . . 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 4274	. . . 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 2318	. 2 |- ( B e. V -> G e. _V )
9		funsng 5376	. . . 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 5347	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. } )
12	10, 11	sylibr 134	. 2 |- ( B e. V -> Fun G )
13		snidg 3698	. . . 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 2325	. 2 |- ( B e. V -> <. ( Base ` ndx ) , B >. e. G )
16	1, 8, 12, 15	strslfvd 13123	1 |- ( B e. V -> B = ( Base ` G ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   {csn 3669   <.cop 3672   Fun wfun 5320   ` cfv 5326   NNcn 9142   ndxcnx 13078   Basecbs 13081

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087

This theorem is referenced by: (None)