Theorem 1strbas 13263

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 13203	. 2 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
2		1str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. }
3		basendxnn 13201	. . . . 5 |- ( Base ` ndx ) e. NN
4		opexg 4326	. . . . 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 4280	. . . 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 5383	. . . 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 5354	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. } )
12	10, 11	sylibr 134	. 2 |- ( B e. V -> Fun G )
13		snidg 3702	. . . 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 13187	1 |- ( B e. V -> B = ( Base ` G ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   {csn 3673   <.cop 3676   Fun wfun 5327   ` cfv 5333   NNcn 9185   ndxcnx 13142   Basecbs 13145

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151

This theorem is referenced by: (None)