Theorem basm 13089

Description: A structure whose base is inhabited is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)

Hypothesis

Ref	Expression
basm.b	|- B = ( Base ` G )

Assertion

Ref	Expression
basm	|- ( A e. B -> E. j j e. G )

Distinct variable group:   j, G

Allowed substitution hints:   A( j)   B( j)

Proof of Theorem basm

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ( A e. B -> A e. B )
2		basm.b	. . . 4 |- B = ( Base ` G )
3		baseid 13081	. . . . 5 |- Base = Slot ( Base ` ndx )
4	2	basmex 13087	. . . . 5 |- ( A e. B -> G e. _V )
5		basendxnn 13083	. . . . . 6 |- ( Base ` ndx ) e. NN
6	5	a1i 9	. . . . 5 |- ( A e. B -> ( Base ` ndx ) e. NN )
7	3, 4, 6	strnfvnd 13047	. . . 4 |- ( A e. B -> ( Base ` G ) = ( G ` ( Base ` ndx ) ) )
8	2, 7	eqtrid 2274	. . 3 |- ( A e. B -> B = ( G ` ( Base ` ndx ) ) )
9	1, 8	eleqtrd 2308	. 2 |- ( A e. B -> A e. ( G ` ( Base ` ndx ) ) )
10		elfvm 5659	. 2 |- ( A e. ( G ` ( Base ` ndx ) ) -> E. j j e. G )
11	9, 10	syl 14	1 |- ( A e. B -> E. j j e. G )

Syntax hints:   -> wi 4   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   ` cfv 5317   NNcn 9106   ndxcnx 13024   Basecbs 13027

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

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 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033

This theorem is referenced by:  relelbasov  13090