Theorem basm 13263

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 ∈ 𝐵 → ∃𝑗 𝑗 ∈ 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 ∈ 𝐵 → 𝐴 ∈ 𝐵)
2		basm.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		baseid 13255	. . . . 5 ⊢ Base = Slot (Base‘ndx)
4	2	basmex 13261	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → 𝐺 ∈ V)
5		basendxnn 13257	. . . . . 6 ⊢ (Base‘ndx) ∈ ℕ
6	5	a1i 9	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (Base‘ndx) ∈ ℕ)
7	3, 4, 6	strnfvnd 13221	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (Base‘𝐺) = (𝐺‘(Base‘ndx)))
8	2, 7	eqtrid 2277	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐵 = (𝐺‘(Base‘ndx)))
9	1, 8	eleqtrd 2311	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐺‘(Base‘ndx)))
10		elfvm 5702	. 2 ⊢ (𝐴 ∈ (𝐺‘(Base‘ndx)) → ∃𝑗 𝑗 ∈ 𝐺)
11	9, 10	syl 14	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑗 𝑗 ∈ 𝐺)

Syntax hints:   → wi 4   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2812  ‘cfv 5351  ℕcn 9233  ndxcnx 13198  Basecbs 13201

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-inn 9234  df-ndx 13204  df-slot 13205  df-base 13207

This theorem is referenced by:  relelbasov  13264