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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 6552 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | df-br 4584 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
3 | ovprc1.1 | . . . . . 6 ⊢ Rel dom 𝐹 | |
4 | brrelex12 5079 | . . . . . 6 ⊢ ((Rel dom 𝐹 ∧ 𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 3, 4 | mpan 702 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
6 | 2, 5 | sylbir 224 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | 6 | con3i 149 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
8 | ndmfv 6128 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘〈𝐴, 𝐵〉) = ∅) |
10 | 1, 9 | syl5eq 2656 | 1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 〈cop 4131 class class class wbr 4583 dom cdm 5038 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: ovprc1 6582 ovprc2 6583 ovrcl 6584 elbasov 15749 firest 15916 psrplusg 19202 psrmulr 19205 psrvscafval 19211 mplval 19249 opsrle 19296 opsrbaslem 19298 opsrbaslemOLD 19299 evlval 19345 matbas0pc 20034 mdetfval 20211 madufval 20262 mdegfval 23626 brovmptimex 37345 nbgrprc0 40555 |
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