Theorem basm 12739

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 12732	. . . . 5 |- Base = Slot ( Base ` ndx )
4	2	basmex 12737	. . . . 5 |- ( A e. B -> G e. _V )
5		basendxnn 12734	. . . . . 6 |- ( Base ` ndx ) e. NN
6	5	a1i 9	. . . . 5 |- ( A e. B -> ( Base ` ndx ) e. NN )
7	3, 4, 6	strnfvnd 12698	. . . 4 |- ( A e. B -> ( Base ` G ) = ( G ` ( Base ` ndx ) ) )
8	2, 7	eqtrid 2241	. . 3 |- ( A e. B -> B = ( G ` ( Base ` ndx ) ) )
9	1, 8	eleqtrd 2275	. 2 |- ( A e. B -> A e. ( G ` ( Base ` ndx ) ) )
10		elfvm 5591	. 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 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   ` cfv 5258   NNcn 8990   ndxcnx 12675   Basecbs 12678

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684

This theorem is referenced by:  relelbasov  12740