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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 7138 | . 2 ⊢ (𝐴⊼𝑔𝐵) = (⊼𝑔‘〈𝐴, 𝐵〉) | |
2 | opelvvg 5559 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
3 | opeq2 4765 | . . . 4 ⊢ (𝑥 = 〈𝐴, 𝐵〉 → 〈1o, 𝑥〉 = 〈1o, 〈𝐴, 𝐵〉〉) | |
4 | df-gona 32701 | . . . 4 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ 〈1o, 𝑥〉) | |
5 | opex 5321 | . . . 4 ⊢ 〈1o, 〈𝐴, 𝐵〉〉 ∈ V | |
6 | 3, 4, 5 | fvmpt 6745 | . . 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 2845 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = 〈1o, 〈𝐴, 𝐵〉〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 × cxp 5517 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ⊼𝑔cgna 32694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-gona 32701 |
This theorem is referenced by: gonanegoal 32712 fmlaomn0 32750 gonan0 32752 gonarlem 32754 gonar 32755 fmla0disjsuc 32758 fmlasucdisj 32759 satffunlem 32761 satffunlem1lem1 32762 satffunlem2lem1 32764 |
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