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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 5770 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | opprc 3721 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
3 | 0ex 4050 | . . . 4 ⊢ ∅ ∈ V | |
4 | 2, 3 | eqeltrdi 2228 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ∈ V) |
5 | df-br 3925 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
6 | ovprc1.1 | . . . . . 6 ⊢ Rel dom 𝐹 | |
7 | brrelex12 4572 | . . . . . 6 ⊢ ((Rel dom 𝐹 ∧ 𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
8 | 6, 7 | mpan 420 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
9 | 5, 8 | sylbir 134 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
10 | 9 | con3i 621 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
11 | ndmfvg 5445 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
12 | 4, 10, 11 | syl2anc 408 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘〈𝐴, 𝐵〉) = ∅) |
13 | 1, 12 | syl5eq 2182 | 1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2681 ∅c0 3358 〈cop 3525 class class class wbr 3924 dom cdm 4534 Rel wrel 4539 ‘cfv 5118 (class class class)co 5767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-dm 4544 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: ovprc1 5800 ovprc2 5801 |
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