Theorem 1strbas 13136

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	⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))

Proof of Theorem 1strbas

Step	Hyp	Ref	Expression
1		baseslid 13076	. 2 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
2		1str.g	. . 3 ⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}
3		basendxnn 13074	. . . . 5 ⊢ (Base‘ndx) ∈ ℕ
4		opexg 4313	. . . . 5 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ 𝑉) → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
5	3, 4	mpan 424	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
6		snexg 4267	. . . 4 ⊢ (⟨(Base‘ndx), 𝐵⟩ ∈ V → {⟨(Base‘ndx), 𝐵⟩} ∈ V)
7	5, 6	syl 14	. . 3 ⊢ (𝐵 ∈ 𝑉 → {⟨(Base‘ndx), 𝐵⟩} ∈ V)
8	2, 7	eqeltrid 2316	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐺 ∈ V)
9		funsng 5363	. . . 4 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ 𝑉) → Fun {⟨(Base‘ndx), 𝐵⟩})
10	3, 9	mpan 424	. . 3 ⊢ (𝐵 ∈ 𝑉 → Fun {⟨(Base‘ndx), 𝐵⟩})
11	2	funeqi 5335	. . 3 ⊢ (Fun 𝐺 ↔ Fun {⟨(Base‘ndx), 𝐵⟩})
12	10, 11	sylibr 134	. 2 ⊢ (𝐵 ∈ 𝑉 → Fun 𝐺)
13		snidg 3695	. . . 4 ⊢ (⟨(Base‘ndx), 𝐵⟩ ∈ V → ⟨(Base‘ndx), 𝐵⟩ ∈ {⟨(Base‘ndx), 𝐵⟩})
14	5, 13	syl 14	. . 3 ⊢ (𝐵 ∈ 𝑉 → ⟨(Base‘ndx), 𝐵⟩ ∈ {⟨(Base‘ndx), 𝐵⟩})
15	14, 2	eleqtrrdi 2323	. 2 ⊢ (𝐵 ∈ 𝑉 → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝐺)
16	1, 8, 12, 15	strslfvd 13060	1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799  {csn 3666  ⟨cop 3669  Fun wfun 5308  ‘cfv 5314  ℕcn 9098  ndxcnx 13015  Basecbs 13018

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fv 5322  df-inn 9099  df-ndx 13021  df-slot 13022  df-base 13024

This theorem is referenced by: (None)