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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 5308 | . . . . 5 ⊢ ∅ ∈ V | |
2 | elmapg 8833 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) | |
3 | 1, 2 | mpan2 690 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓:∅⟶𝐵)) |
4 | f0bi 6775 | . . . 4 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅) | |
5 | 3, 4 | bitrdi 287 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 = ∅)) |
6 | velsn 4645 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
7 | 5, 6 | bitr4di 289 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑m ∅) ↔ 𝑓 ∈ {∅})) |
8 | 7 | eqrdv 2731 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑m ∅) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4323 {csn 4629 ⟶wf 6540 (class class class)co 7409 ↑m cmap 8820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-map 8822 |
This theorem is referenced by: map0e 8876 hashmap 14395 ehl0base 24933 repr0 33654 mpct 43948 rrxtopn0 45057 qndenserrnbl 45059 hoicvr 45312 ovn02 45332 ovnhoi 45367 ovnlecvr2 45374 hoiqssbl 45389 hoimbl 45395 0aryfvalel 47368 |
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