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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 5175 | . . . . 5 ⊢ ∅ ∈ V | |
2 | elmapg 8402 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) | |
3 | 1, 2 | mpan2 690 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) |
4 | f0bi 6536 | . . . 4 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅) | |
5 | 3, 4 | syl6bb 290 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 = ∅)) |
6 | velsn 4541 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
7 | 5, 6 | syl6bbr 292 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 ∈ {∅})) |
8 | 7 | eqrdv 2796 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑m ∅) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 ⟶wf 6320 (class class class)co 7135 ↑m cmap 8389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 |
This theorem is referenced by: map0e 8429 hashmap 13792 ehl0base 24020 repr0 31992 mpct 41830 rrxtopn0 42935 qndenserrnbl 42937 hoicvr 43187 ovn02 43207 ovnhoi 43242 ovnlecvr2 43249 hoiqssbl 43264 hoimbl 43270 0aryfvalel 45048 |
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