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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 7152 | . 2 ⊢ (𝐴⊼𝑔𝐵) = (⊼𝑔‘〈𝐴, 𝐵〉) | |
2 | opelvvg 5588 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
3 | opeq2 4797 | . . . 4 ⊢ (𝑥 = 〈𝐴, 𝐵〉 → 〈1o, 𝑥〉 = 〈1o, 〈𝐴, 𝐵〉〉) | |
4 | df-gona 32607 | . . . 4 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ 〈1o, 𝑥〉) | |
5 | opex 5349 | . . . 4 ⊢ 〈1o, 〈𝐴, 𝐵〉〉 ∈ V | |
6 | 3, 4, 5 | fvmpt 6761 | . . 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 2867 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = 〈1o, 〈𝐴, 𝐵〉〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 〈cop 4566 × cxp 5546 ‘cfv 6348 (class class class)co 7149 1oc1o 8088 ⊼𝑔cgna 32600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-gona 32607 |
This theorem is referenced by: gonanegoal 32618 fmlaomn0 32656 gonan0 32658 gonarlem 32660 gonar 32661 fmla0disjsuc 32664 fmlasucdisj 32665 satffunlem 32667 satffunlem1lem1 32668 satffunlem2lem1 32670 |
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