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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmaov | Structured version Visualization version GIF version |
Description: The value of an operation outside its domain, analogous to ndmafv 43332. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
ndmaov | ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aov 43313 | . 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
2 | ndmafv 43332 | . 2 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐹'''〈𝐴, 𝐵〉) = V) | |
3 | 1, 2 | syl5eq 2868 | 1 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3495 〈cop 4567 dom cdm 5550 '''cafv 43309 ((caov 43310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-int 4870 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-res 5562 df-iota 6309 df-fun 6352 df-fv 6358 df-aiota 43278 df-dfat 43311 df-afv 43312 df-aov 43313 |
This theorem is referenced by: ndmaovg 43376 ndmaovcl 43395 ndmaovcom 43397 ndmaovass 43398 ndmaovdistr 43399 |
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