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Mirrors > Home > MPE Home > Th. List > ovima0 | Structured version Visualization version GIF version |
Description: An operation value is a member of the image plus null. (Contributed by Thierry Arnoux, 25-Jun-2019.) |
Ref | Expression |
---|---|
ovima0 | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) = ∅) | |
2 | ssun2 4171 | . . . 4 ⊢ {∅} ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) | |
3 | 0ex 5308 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | snid 4666 | . . . 4 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3973 | . . 3 ⊢ ∅ ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) |
6 | 1, 5 | eqeltrdi 2833 | . 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
7 | ssun1 4170 | . . 3 ⊢ (𝑅 “ (𝐴 × 𝐵)) ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) | |
8 | df-ov 7422 | . . . 4 ⊢ (𝑋𝑅𝑌) = (𝑅‘〈𝑋, 𝑌〉) | |
9 | opelxpi 5715 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) | |
10 | 8 | eqeq1i 2730 | . . . . . . 7 ⊢ ((𝑋𝑅𝑌) = ∅ ↔ (𝑅‘〈𝑋, 𝑌〉) = ∅) |
11 | 10 | notbii 319 | . . . . . 6 ⊢ (¬ (𝑋𝑅𝑌) = ∅ ↔ ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) |
12 | 11 | biimpi 215 | . . . . 5 ⊢ (¬ (𝑋𝑅𝑌) = ∅ → ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) |
13 | eliman0 6936 | . . . . 5 ⊢ ((〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵) ∧ ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) → (𝑅‘〈𝑋, 𝑌〉) ∈ (𝑅 “ (𝐴 × 𝐵))) | |
14 | 9, 12, 13 | syl2an 594 | . . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑅‘〈𝑋, 𝑌〉) ∈ (𝑅 “ (𝐴 × 𝐵))) |
15 | 8, 14 | eqeltrid 2829 | . . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ (𝑅 “ (𝐴 × 𝐵))) |
16 | 7, 15 | sselid 3974 | . 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
17 | 6, 16 | pm2.61dan 811 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3942 ∅c0 4322 {csn 4630 〈cop 4636 × cxp 5676 “ cima 5681 ‘cfv 6549 (class class class)co 7419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fv 6557 df-ov 7422 |
This theorem is referenced by: legval 28465 |
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