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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 5313 | . . . . 5 ⊢ ∅ ∈ V | |
2 | elmapg 8878 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) | |
3 | 1, 2 | mpan2 691 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) |
4 | f0bi 6792 | . . . 4 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅) | |
5 | 3, 4 | bitrdi 287 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 = ∅)) |
6 | velsn 4647 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
7 | 5, 6 | bitr4di 289 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 ∈ {∅})) |
8 | 7 | eqrdv 2733 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑m ∅) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 ⟶wf 6559 (class class class)co 7431 ↑m cmap 8865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 |
This theorem is referenced by: map0e 8921 hashmap 14471 ehl0base 25464 1arithidom 33545 repr0 34605 mpct 45144 rrxtopn0 46249 qndenserrnbl 46251 hoicvr 46504 ovn02 46524 ovnhoi 46559 ovnlecvr2 46566 hoiqssbl 46581 hoimbl 46587 0aryfvalel 48484 |
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