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) ∈ ℕ)

Assertion

Ref	Expression
slotm	⊢ (𝐴 ∈ (𝐸‘𝐺) → ∃𝑗 𝑗 ∈ 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‘𝐺) → 𝐴 ∈ (𝐸‘𝐺))
2		slotm.e	. . . . 5 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
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‘𝐺)) → 𝐺 ∈ dom 𝐸)
8	6, 7	mpan 428	. . . 4 ⊢ (𝐴 ∈ (𝐸‘𝐺) → 𝐺 ∈ dom 𝐸)
9	2	simpri 113	. . . . 5 ⊢ (𝐸‘ndx) ∈ ℕ
10	9	a1i 9	. . . 4 ⊢ (𝐴 ∈ (𝐸‘𝐺) → (𝐸‘ndx) ∈ ℕ)
11	3, 8, 10	strnfvnd 13355	. . 3 ⊢ (𝐴 ∈ (𝐸‘𝐺) → (𝐸‘𝐺) = (𝐺‘(𝐸‘ndx)))
12	1, 11	eleqtrd 2317	. 2 ⊢ (𝐴 ∈ (𝐸‘𝐺) → 𝐴 ∈ (𝐺‘(𝐸‘ndx)))
13		elfvm 5726	. 2 ⊢ (𝐴 ∈ (𝐺‘(𝐸‘ndx)) → ∃𝑗 𝑗 ∈ 𝐺)
14	12, 13	syl 14	1 ⊢ (𝐴 ∈ (𝐸‘𝐺) → ∃𝑗 𝑗 ∈ 𝐺)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  dom cdm 4772  Rel wrel 4777   Fn wfn 5370  ‘cfv 5375  ℕcn 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