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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 477 | . . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) = ∅) | |
2 | ssun2 3777 | . . . 4 ⊢ {∅} ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) | |
3 | 0ex 4790 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | snid 4208 | . . . 4 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3600 | . . 3 ⊢ ∅ ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) |
6 | 1, 5 | syl6eqel 2709 | . 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
7 | ssun1 3776 | . . 3 ⊢ (𝑅 “ (𝐴 × 𝐵)) ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}) | |
8 | df-ov 6653 | . . . 4 ⊢ (𝑋𝑅𝑌) = (𝑅‘〈𝑋, 𝑌〉) | |
9 | opelxpi 5148 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) | |
10 | 8 | eqeq1i 2627 | . . . . . . 7 ⊢ ((𝑋𝑅𝑌) = ∅ ↔ (𝑅‘〈𝑋, 𝑌〉) = ∅) |
11 | 10 | notbii 310 | . . . . . 6 ⊢ (¬ (𝑋𝑅𝑌) = ∅ ↔ ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) |
12 | 11 | biimpi 206 | . . . . 5 ⊢ (¬ (𝑋𝑅𝑌) = ∅ → ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) |
13 | eliman0 6223 | . . . . 5 ⊢ ((〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵) ∧ ¬ (𝑅‘〈𝑋, 𝑌〉) = ∅) → (𝑅‘〈𝑋, 𝑌〉) ∈ (𝑅 “ (𝐴 × 𝐵))) | |
14 | 9, 12, 13 | syl2an 494 | . . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑅‘〈𝑋, 𝑌〉) ∈ (𝑅 “ (𝐴 × 𝐵))) |
15 | 8, 14 | syl5eqel 2705 | . . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ (𝑅 “ (𝐴 × 𝐵))) |
16 | 7, 15 | sseldi 3601 | . 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
17 | 6, 16 | pm2.61dan 832 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ∅c0 3915 {csn 4177 〈cop 4183 × cxp 5112 “ cima 5117 ‘cfv 5888 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: legval 25479 |
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