Theorem basm 13143

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 13135	. . . . 5 |- Base = Slot ( Base ` ndx )
4	2	basmex 13141	. . . . 5 |- ( A e. B -> G e. _V )
5		basendxnn 13137	. . . . . 6 |- ( Base ` ndx ) e. NN
6	5	a1i 9	. . . . 5 |- ( A e. B -> ( Base ` ndx ) e. NN )
7	3, 4, 6	strnfvnd 13101	. . . 4 |- ( A e. B -> ( Base ` G ) = ( G ` ( Base ` ndx ) ) )
8	2, 7	eqtrid 2276	. . 3 |- ( A e. B -> B = ( G ` ( Base ` ndx ) ) )
9	1, 8	eleqtrd 2310	. 2 |- ( A e. B -> A e. ( G ` ( Base ` ndx ) ) )
10		elfvm 5672	. 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 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   ` 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-fn 5329  df-fv 5334  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087

This theorem is referenced by:  relelbasov  13144