ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1strbas GIF version

Theorem 1strbas 12568
Description: The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
Hypothesis
Ref Expression
1str.g 𝐺 = {⟨(Base‘ndx), 𝐵⟩}
Assertion
Ref Expression
1strbas (𝐵𝑉𝐵 = (Base‘𝐺))

Proof of Theorem 1strbas
StepHypRef Expression
1 baseslid 12511 . 2 (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
2 1str.g . . 3 𝐺 = {⟨(Base‘ndx), 𝐵⟩}
3 basendxnn 12510 . . . . 5 (Base‘ndx) ∈ ℕ
4 opexg 4227 . . . . 5 (((Base‘ndx) ∈ ℕ ∧ 𝐵𝑉) → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
53, 4mpan 424 . . . 4 (𝐵𝑉 → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
6 snexg 4183 . . . 4 (⟨(Base‘ndx), 𝐵⟩ ∈ V → {⟨(Base‘ndx), 𝐵⟩} ∈ V)
75, 6syl 14 . . 3 (𝐵𝑉 → {⟨(Base‘ndx), 𝐵⟩} ∈ V)
82, 7eqeltrid 2264 . 2 (𝐵𝑉𝐺 ∈ V)
9 funsng 5261 . . . 4 (((Base‘ndx) ∈ ℕ ∧ 𝐵𝑉) → Fun {⟨(Base‘ndx), 𝐵⟩})
103, 9mpan 424 . . 3 (𝐵𝑉 → Fun {⟨(Base‘ndx), 𝐵⟩})
112funeqi 5236 . . 3 (Fun 𝐺 ↔ Fun {⟨(Base‘ndx), 𝐵⟩})
1210, 11sylibr 134 . 2 (𝐵𝑉 → Fun 𝐺)
13 snidg 3621 . . . 4 (⟨(Base‘ndx), 𝐵⟩ ∈ V → ⟨(Base‘ndx), 𝐵⟩ ∈ {⟨(Base‘ndx), 𝐵⟩})
145, 13syl 14 . . 3 (𝐵𝑉 → ⟨(Base‘ndx), 𝐵⟩ ∈ {⟨(Base‘ndx), 𝐵⟩})
1514, 2eleqtrrdi 2271 . 2 (𝐵𝑉 → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝐺)
161, 8, 12, 15strslfvd 12496 1 (𝐵𝑉𝐵 = (Base‘𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737  {csn 3592  cop 3595  Fun wfun 5209  cfv 5215  cn 8915  ndxcnx 12451  Basecbs 12454
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fv 5223  df-inn 8916  df-ndx 12457  df-slot 12458  df-base 12460
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator