![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 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 ∧ 𝐵 ∈ 𝑉) → 〈(Base‘ndx), 𝐵〉 ∈ V) | |
5 | 3, 4 | mpan 424 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → 〈(Base‘ndx), 𝐵〉 ∈ V) |
6 | snexg 4183 | . . . 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 5261 | . . . 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 5236 | . . 3 ⊢ (Fun 𝐺 ↔ Fun {〈(Base‘ndx), 𝐵〉}) |
12 | 10, 11 | sylibr 134 | . 2 ⊢ (𝐵 ∈ 𝑉 → Fun 𝐺) |
13 | snidg 3621 | . . . 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 12496 | 1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (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 ℕcn 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 |