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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 261 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ 𝐴 ≠ ∅) | |
| 4 | 3 | necon2bbii 2992 | . . . . . 6 ⊢ (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅) | 
| 5 | 4 | imbi2i 336 | . . . . 5 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐵 = ∅ → ¬ 𝐴 ≠ ∅)) | 
| 6 | imnan 399 | . . . . 5 ⊢ ((𝐵 = ∅ → ¬ 𝐴 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅)) | |
| 7 | 5, 6 | bitri 275 | . . . 4 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐴 ≠ ∅)) | 
| 8 | map0g 8924 | . . . . 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 4352 | . . . 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 8879 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵)) | |
| 14 | 13 | exbidv 1921 | . . 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 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 ⟶wf 6557 (class class class)co 7431 ↑m cmap 8866 | 
| 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-pow 5365 ax-pr 5432 ax-un 7755 | 
| 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-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 | 
| This theorem is referenced by: (None) | 
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