Theorem slotm 13398

Description: A structure with an inhabited slot is inhabited. (Contributed by Jim Kingdon, 24-Jul-2026.)

Hypothesis

Ref	Expression
slotm.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )

Assertion

Ref	Expression
slotm	|- ( A e. ( E ` G ) -> E. j j e. G )

Distinct variable group:   j, G

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

Proof of Theorem slotm

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ( A e. ( E ` G ) -> A e. ( E ` G ) )
2		slotm.e	. . . . 5 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
3	2	simpli 111	. . . 4 |- E = Slot ( E ` ndx )
4	2	slotslfn 13361	. . . . . 6 |- E Fn _V
5		fnrel 5477	. . . . . 6 |- ( E Fn _V -> Rel E )
6	4, 5	ax-mp 5	. . . . 5 |- Rel E
7		relelfvdm 5725	. . . . 5 |- ( ( Rel E /\ A e. ( E ` G ) ) -> G e. dom E )
8	6, 7	mpan 428	. . . 4 |- ( A e. ( E ` G ) -> G e. dom E )
9	2	simpri 113	. . . . 5 |- ( E ` ndx ) e. NN
10	9	a1i 9	. . . 4 |- ( A e. ( E ` G ) -> ( E ` ndx ) e. NN )
11	3, 8, 10	strnfvnd 13355	. . 3 |- ( A e. ( E ` G ) -> ( E ` G ) = ( G ` ( E ` ndx ) ) )
12	1, 11	eleqtrd 2317	. 2 |- ( A e. ( E ` G ) -> A e. ( G ` ( E ` ndx ) ) )
13		elfvm 5726	. 2 |- ( A e. ( G ` ( E ` ndx ) ) -> E. j j e. G )
14	12, 13	syl 14	1 |- ( A e. ( E ` G ) -> E. j j e. G )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   dom cdm 4772   Rel wrel 4777   Fn wfn 5370   ` cfv 5375   NNcn 9287   ndxcnx 13332  Slot cslot 13334

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-slot 13339

This theorem is referenced by:  mgpplusg  14205