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| Mirrors > Home > MPE Home > Th. List > mapdm0 | Structured version Visualization version GIF version | ||
| Description: The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.) |
| Ref | Expression |
|---|---|
| mapdm0 | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑m ∅) = {∅}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 2 | elmapg 8879 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) | |
| 3 | 1, 2 | mpan2 691 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) |
| 4 | f0bi 6791 | . . . 4 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅) | |
| 5 | 3, 4 | bitrdi 287 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 = ∅)) |
| 6 | velsn 4642 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
| 7 | 5, 6 | bitr4di 289 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 ∈ {∅})) |
| 8 | 7 | eqrdv 2735 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑m ∅) = {∅}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 ⟶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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 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-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 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-map 8868 |
| This theorem is referenced by: map0e 8922 hashmap 14474 ehl0base 25450 1arithidom 33565 repr0 34626 mpct 45206 rrxtopn0 46308 qndenserrnbl 46310 hoicvr 46563 ovn02 46583 ovnhoi 46618 ovnlecvr2 46625 hoiqssbl 46640 hoimbl 46646 0aryfvalel 48555 |
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