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Mirrors > Home > ILE Home > Th. List > ovprc | 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 5877 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | opprc 3799 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
3 | 0ex 4130 | . . . 4 ⊢ ∅ ∈ V | |
4 | 2, 3 | eqeltrdi 2268 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ∈ V) |
5 | df-br 4004 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
6 | ovprc1.1 | . . . . . 6 ⊢ Rel dom 𝐹 | |
7 | brrelex12 4664 | . . . . . 6 ⊢ ((Rel dom 𝐹 ∧ 𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
8 | 6, 7 | mpan 424 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
9 | 5, 8 | sylbir 135 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
10 | 9 | con3i 632 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
11 | ndmfvg 5546 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
12 | 4, 10, 11 | syl2anc 411 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘〈𝐴, 𝐵〉) = ∅) |
13 | 1, 12 | eqtrid 2222 | 1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∅c0 3422 〈cop 3595 class class class wbr 4003 dom cdm 4626 Rel wrel 4631 ‘cfv 5216 (class class class)co 5874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-rel 4633 df-dm 4636 df-iota 5178 df-fv 5224 df-ov 5877 |
This theorem is referenced by: ovprc1 5910 ovprc2 5911 |
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