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Mirrors > Home > MPE Home > Th. List > Mathboxes > map0cor | Structured version Visualization version GIF version |
Description: A function exists iff an empty codomain is accompanied with an empty domain. (Contributed by Zhi Wang, 1-Oct-2024.) |
Ref | Expression |
---|---|
map0cor.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
map0cor.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
map0cor | ⊢ (𝜑 → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴⟶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | map0cor.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
2 | map0cor.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | biid 260 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ 𝐴 ≠ ∅) | |
4 | 3 | necon2bbii 2995 | . . . . . 6 ⊢ (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅) |
5 | 4 | imbi2i 336 | . . . . 5 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐵 = ∅ → ¬ 𝐴 ≠ ∅)) |
6 | imnan 400 | . . . . 5 ⊢ ((𝐵 = ∅ → ¬ 𝐴 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅)) | |
7 | 5, 6 | bitri 274 | . . . 4 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅)) |
8 | map0g 8661 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((𝐵 ↑m 𝐴) = ∅ ↔ (𝐵 = ∅ ∧ 𝐴 ≠ ∅))) | |
9 | 8 | notbid 318 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (¬ (𝐵 ↑m 𝐴) = ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅))) |
10 | 7, 9 | bitr4id 290 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ¬ (𝐵 ↑m 𝐴) = ∅)) |
11 | neq0 4281 | . . . 4 ⊢ (¬ (𝐵 ↑m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵 ↑m 𝐴)) | |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (¬ (𝐵 ↑m 𝐴) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝐵 ↑m 𝐴))) |
13 | elmapg 8617 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵)) | |
14 | 13 | exbidv 1924 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (∃𝑓 𝑓 ∈ (𝐵 ↑m 𝐴) ↔ ∃𝑓 𝑓:𝐴⟶𝐵)) |
15 | 10, 12, 14 | 3bitrd 305 | . 2 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴⟶𝐵)) |
16 | 1, 2, 15 | syl2anc 584 | 1 ⊢ (𝜑 → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴⟶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∅c0 4258 ⟶wf 6424 (class class class)co 7269 ↑m cmap 8604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5486 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-fv 6436 df-ov 7272 df-oprab 7273 df-mpo 7274 df-1st 7822 df-2nd 7823 df-map 8606 |
This theorem is referenced by: (None) |
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