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Mirrors > Home > ILE Home > Th. List > 1strbas | GIF version |
Description: The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) |
Ref | Expression |
---|---|
1str.g | ⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩} |
Ref | Expression |
---|---|
1strbas | ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseslid 12522 | . 2 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ) | |
2 | 1str.g | . . 3 ⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩} | |
3 | basendxnn 12521 | . . . . 5 ⊢ (Base‘ndx) ∈ ℕ | |
4 | opexg 4230 | . . . . 5 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ 𝑉) → ⟨(Base‘ndx), 𝐵⟩ ∈ V) | |
5 | 3, 4 | mpan 424 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ⟨(Base‘ndx), 𝐵⟩ ∈ V) |
6 | snexg 4186 | . . . 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 2264 | . 2 ⊢ (𝐵 ∈ 𝑉 → 𝐺 ∈ V) |
9 | funsng 5264 | . . . 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 5239 | . . 3 ⊢ (Fun 𝐺 ↔ Fun {⟨(Base‘ndx), 𝐵⟩}) |
12 | 10, 11 | sylibr 134 | . 2 ⊢ (𝐵 ∈ 𝑉 → Fun 𝐺) |
13 | snidg 3623 | . . . 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 2271 | . 2 ⊢ (𝐵 ∈ 𝑉 → ⟨(Base‘ndx), 𝐵⟩ ∈ 𝐺) |
16 | 1, 8, 12, 15 | strslfvd 12507 | 1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2739 {csn 3594 ⟨cop 3597 Fun wfun 5212 ‘cfv 5218 ℕcn 8922 ndxcnx 12462 Basecbs 12465 |
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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7905 ax-resscn 7906 ax-1re 7908 ax-addrcl 7911 |
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 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-iota 5180 df-fun 5220 df-fv 5226 df-inn 8923 df-ndx 12468 df-slot 12469 df-base 12471 |
This theorem is referenced by: (None) |
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