Theorem basm 13207

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 13199	. . . . 5 |- Base = Slot ( Base ` ndx )
4	2	basmex 13205	. . . . 5 |- ( A e. B -> G e. _V )
5		basendxnn 13201	. . . . . 6 |- ( Base ` ndx ) e. NN
6	5	a1i 9	. . . . 5 |- ( A e. B -> ( Base ` ndx ) e. NN )
7	3, 4, 6	strnfvnd 13165	. . . 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 5681	. 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 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   ` 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-fn 5336  df-fv 5341  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151

This theorem is referenced by:  relelbasov  13208