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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 488 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
2 | ovprc1.1 | . . 3 ⊢ Rel dom 𝐹 | |
3 | 2 | ovprc 7173 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
4 | 1, 3 | nsyl5 162 | 1 ⊢ (¬ 𝐵 ∈ V → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 dom cdm 5519 Rel wrel 5524 (class class class)co 7135 |
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-pow 5231 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-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-xp 5525 df-rel 5526 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: ressbasss 16548 ress0 16550 wunress 16556 0rest 16695 firest 16698 subcmn 18950 dprdval0prc 19117 zrhval 20201 dsmmval2 20425 psrbas 20616 psr1val 20815 vr1val 20821 ply1ascl 20887 evl1fval 20952 restbas 21763 resstopn 21791 deg1fval 24681 wwlksn 27623 submomnd 30761 suborng 30939 bj-restsnid 34502 1aryenef 45059 2aryenef 45070 |
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