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Mirrors > Home > MPE Home > Th. List > Mathboxes > gonafv | Structured version Visualization version GIF version |
Description: The "Godel-set for the Sheffer stroke NAND" for two formulas 𝐴 and 𝐵. (Contributed by AV, 16-Oct-2023.) |
Ref | Expression |
---|---|
gonafv | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = 〈1o, 〈𝐴, 𝐵〉〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7235 | . 2 ⊢ (𝐴⊼𝑔𝐵) = (⊼𝑔‘〈𝐴, 𝐵〉) | |
2 | opelvvg 5606 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
3 | opeq2 4800 | . . . 4 ⊢ (𝑥 = 〈𝐴, 𝐵〉 → 〈1o, 𝑥〉 = 〈1o, 〈𝐴, 𝐵〉〉) | |
4 | df-gona 33039 | . . . 4 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ 〈1o, 𝑥〉) | |
5 | opex 5363 | . . . 4 ⊢ 〈1o, 〈𝐴, 𝐵〉〉 ∈ V | |
6 | 3, 4, 5 | fvmpt 6837 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (V × V) → (⊼𝑔‘〈𝐴, 𝐵〉) = 〈1o, 〈𝐴, 𝐵〉〉) |
7 | 2, 6 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⊼𝑔‘〈𝐴, 𝐵〉) = 〈1o, 〈𝐴, 𝐵〉〉) |
8 | 1, 7 | syl5eq 2791 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = 〈1o, 〈𝐴, 𝐵〉〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 Vcvv 3421 〈cop 4562 × cxp 5564 ‘cfv 6398 (class class class)co 7232 1oc1o 8216 ⊼𝑔cgna 33032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-iota 6356 df-fun 6400 df-fv 6406 df-ov 7235 df-gona 33039 |
This theorem is referenced by: gonanegoal 33050 fmlaomn0 33088 gonan0 33090 gonarlem 33092 gonar 33093 fmla0disjsuc 33096 fmlasucdisj 33097 satffunlem 33099 satffunlem1lem1 33100 satffunlem2lem1 33102 |
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