![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 479 | . . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) = ∅) | |
2 | ssun2 4000 | . . . 4 ⊢ {∅} ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) | |
3 | 0ex 5026 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | snid 4430 | . . . 4 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3818 | . . 3 ⊢ ∅ ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) |
6 | 1, 5 | syl6eqel 2867 | . 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
7 | ssun1 3999 | . . 3 ⊢ (𝑅 “ (𝐴 × 𝐵)) ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) | |
8 | df-ov 6925 | . . . 4 ⊢ (𝑋𝑅𝑌) = (𝑅‘〈𝑋, 𝑌〉) | |
9 | opelxpi 5392 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) | |
10 | 8 | eqeq1i 2783 | . . . . . . 7 ⊢ ((𝑋𝑅𝑌) = ∅ ↔ (𝑅‘〈𝑋, 𝑌〉) = ∅) |
11 | 10 | notbii 312 | . . . . . 6 ⊢ (¬ (𝑋𝑅𝑌) = ∅ ↔ ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) |
12 | 11 | biimpi 208 | . . . . 5 ⊢ (¬ (𝑋𝑅𝑌) = ∅ → ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) |
13 | eliman0 6482 | . . . . 5 ⊢ ((〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵) ∧ ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) → (𝑅‘〈𝑋, 𝑌〉) ∈ (𝑅 “ (𝐴 × 𝐵))) | |
14 | 9, 12, 13 | syl2an 589 | . . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑅‘〈𝑋, 𝑌〉) ∈ (𝑅 “ (𝐴 × 𝐵))) |
15 | 8, 14 | syl5eqel 2863 | . . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ (𝑅 “ (𝐴 × 𝐵))) |
16 | 7, 15 | sseldi 3819 | . 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
17 | 6, 16 | pm2.61dan 803 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∪ cun 3790 ∅c0 4141 {csn 4398 〈cop 4404 × cxp 5353 “ cima 5358 ‘cfv 6135 (class class class)co 6922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-xp 5361 df-cnv 5363 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fv 6143 df-ov 6925 |
This theorem is referenced by: legval 25935 |
Copyright terms: Public domain | W3C validator |