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Mirrors > Home > MPE Home > Th. List > ovprc2 | Structured version Visualization version GIF version |
Description: The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
ovprc1.1 | ⊢ Rel dom 𝐹 |
Ref | Expression |
---|---|
ovprc2 | ⊢ (¬ 𝐵 ∈ V → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
2 | ovprc1.1 | . . 3 ⊢ Rel dom 𝐹 | |
3 | 2 | ovprc 7457 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
4 | 1, 3 | nsyl5 159 | 1 ⊢ (¬ 𝐵 ∈ V → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∅c0 4322 dom cdm 5678 Rel wrel 5683 (class class class)co 7419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-dm 5688 df-iota 6501 df-fv 6557 df-ov 7422 |
This theorem is referenced by: ressbasssg 17220 ressbasssOLD 17223 ress0 17227 wunress 17234 wunressOLD 17235 0rest 17414 firest 17417 subcmn 19804 dprdval0prc 19971 zrhval 21450 dsmmval2 21687 psrbas 21895 psr1val 22128 vr1val 22134 ply1ascl 22202 evl1fval 22272 restbas 23106 resstopn 23134 deg1fval 26060 wwlksn 29720 submomnd 32880 suborng 33129 bj-restsnid 36694 1aryenef 47901 2aryenef 47912 |
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