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Mirrors > Home > MPE Home > Th. List > ovprc | Structured version Visualization version GIF version |
Description: The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
ovprc1.1 | ⊢ Rel dom 𝐹 |
Ref | Expression |
---|---|
ovprc | ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 6879 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | df-br 4842 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
3 | ovprc1.1 | . . . . . 6 ⊢ Rel dom 𝐹 | |
4 | 3 | brrelex12i 5360 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | 2, 4 | sylbir 227 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
6 | 5 | con3i 152 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
7 | ndmfv 6439 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘〈𝐴, 𝐵〉) = ∅) |
9 | 1, 8 | syl5eq 2843 | 1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3383 ∅c0 4113 〈cop 4372 class class class wbr 4841 dom cdm 5310 Rel wrel 5315 ‘cfv 6099 (class class class)co 6876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-xp 5316 df-rel 5317 df-dm 5320 df-iota 6062 df-fv 6107 df-ov 6879 |
This theorem is referenced by: ovprc1 6914 ovprc2 6915 ovrcl 6916 elbasov 16243 firest 16405 psrplusg 19701 psrmulr 19704 psrvscafval 19710 mplval 19748 opsrle 19795 opsrbaslem 19797 evlval 19843 matbas0pc 20537 mdetfval 20715 madufval 20766 mdegfval 24160 nbgrprc0 26560 brovmptimex 39095 |
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