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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 7138 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | df-br 5031 | . . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
3 | ovprc1.1 | . . . . 5 ⊢ Rel dom 𝐹 | |
4 | 3 | brrelex12i 5571 | . . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | 2, 4 | sylbir 238 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
6 | ndmfv 6675 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
7 | 5, 6 | nsyl5 162 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘〈𝐴, 𝐵〉) = ∅) |
8 | 1, 7 | syl5eq 2845 | 1 ⊢ (¬ (𝐴 ∈ V ∧ 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 〈cop 4531 class class class wbr 5030 dom cdm 5519 Rel wrel 5524 ‘cfv 6324 (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: ovprc1 7174 ovprc2 7175 ovrcl 7176 elbasov 16537 firest 16698 psrplusg 20619 psrmulr 20622 psrvscafval 20628 mplval 20666 opsrle 20715 opsrbaslem 20717 evlval 20767 matbas0pc 21014 mdetfval 21191 madufval 21242 mdegfval 24663 nbgrprc0 27124 gonan0 32752 brovmptimex 40730 |
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