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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 484 | . . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) = ∅) | |
| 2 | ssun2 4179 | . . . 4 ⊢ {∅} ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) | |
| 3 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 4 | 3 | snid 4662 | . . . 4 ⊢ ∅ ∈ {∅} | 
| 5 | 2, 4 | sselii 3980 | . . 3 ⊢ ∅ ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) | 
| 6 | 1, 5 | eqeltrdi 2849 | . 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) | 
| 7 | ssun1 4178 | . . 3 ⊢ (𝑅 “ (𝐴 × 𝐵)) ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) | |
| 8 | df-ov 7434 | . . . 4 ⊢ (𝑋𝑅𝑌) = (𝑅‘〈𝑋, 𝑌〉) | |
| 9 | opelxpi 5722 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) | |
| 10 | 8 | eqeq1i 2742 | . . . . . . 7 ⊢ ((𝑋𝑅𝑌) = ∅ ↔ (𝑅‘〈𝑋, 𝑌〉) = ∅) | 
| 11 | 10 | notbii 320 | . . . . . 6 ⊢ (¬ (𝑋𝑅𝑌) = ∅ ↔ ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) | 
| 12 | 11 | biimpi 216 | . . . . 5 ⊢ (¬ (𝑋𝑅𝑌) = ∅ → ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) | 
| 13 | eliman0 6946 | . . . . 5 ⊢ ((〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵) ∧ ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) → (𝑅‘〈𝑋, 𝑌〉) ∈ (𝑅 “ (𝐴 × 𝐵))) | |
| 14 | 9, 12, 13 | syl2an 596 | . . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑅‘〈𝑋, 𝑌〉) ∈ (𝑅 “ (𝐴 × 𝐵))) | 
| 15 | 8, 14 | eqeltrid 2845 | . . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ (𝑅 “ (𝐴 × 𝐵))) | 
| 16 | 7, 15 | sselid 3981 | . 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) | 
| 17 | 6, 16 | pm2.61dan 813 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∅c0 4333 {csn 4626 〈cop 4632 × cxp 5683 “ cima 5688 ‘cfv 6561 (class class class)co 7431 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: legval 28592 | 
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