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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 5214 | . . . . 5 ⊢ ∅ ∈ V | |
2 | elmapg 8422 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) | |
3 | 1, 2 | mpan2 689 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) |
4 | f0bi 6565 | . . . 4 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅) | |
5 | 3, 4 | syl6bb 289 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 = ∅)) |
6 | velsn 4586 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
7 | 5, 6 | syl6bbr 291 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 ∈ {∅})) |
8 | 7 | eqrdv 2822 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑m ∅) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 {csn 4570 ⟶wf 6354 (class class class)co 7159 ↑m cmap 8409 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 |
This theorem is referenced by: map0e 8449 hashmap 13799 ehl0base 24022 repr0 31886 mpct 41470 rrxtopn0 42585 qndenserrnbl 42587 hoicvr 42837 ovn02 42857 ovnhoi 42892 ovnlecvr2 42899 hoiqssbl 42914 hoimbl 42920 |
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