Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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.) |
Ref | Expression |
---|---|
mapdm0 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4718 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | elmapg 7757 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴)) | |
3 | 1, 2 | mpan2 703 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴)) |
4 | 3 | biimpa 500 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (𝐴 ↑𝑚 ∅)) → 𝑓:∅⟶𝐴) |
5 | f0bi 6001 | . . . 4 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 = ∅) | |
6 | 4, 5 | sylib 207 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (𝐴 ↑𝑚 ∅)) → 𝑓 = ∅) |
7 | 6 | ralrimiva 2949 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑓 ∈ (𝐴 ↑𝑚 ∅)𝑓 = ∅) |
8 | f0 5999 | . . . . . 6 ⊢ ∅:∅⟶𝐴 | |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∅:∅⟶𝐴) |
10 | id 22 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
11 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V) |
12 | elmapg 7757 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (∅ ∈ (𝐴 ↑𝑚 ∅) ↔ ∅:∅⟶𝐴)) | |
13 | 10, 11, 12 | syl2anc 691 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∅ ∈ (𝐴 ↑𝑚 ∅) ↔ ∅:∅⟶𝐴)) |
14 | 9, 13 | mpbird 246 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ (𝐴 ↑𝑚 ∅)) |
15 | ne0i 3880 | . . . 4 ⊢ (∅ ∈ (𝐴 ↑𝑚 ∅) → (𝐴 ↑𝑚 ∅) ≠ ∅) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) ≠ ∅) |
17 | eqsn 4301 | . . 3 ⊢ ((𝐴 ↑𝑚 ∅) ≠ ∅ → ((𝐴 ↑𝑚 ∅) = {∅} ↔ ∀𝑓 ∈ (𝐴 ↑𝑚 ∅)𝑓 = ∅)) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑𝑚 ∅) = {∅} ↔ ∀𝑓 ∈ (𝐴 ↑𝑚 ∅)𝑓 = ∅)) |
19 | 7, 18 | mpbird 246 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∅c0 3874 {csn 4125 ⟶wf 5800 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 |
This theorem is referenced by: mpct 38388 rrxtopn0 39189 qndenserrnbl 39191 hoicvr 39438 ovn02 39458 ovnhoi 39493 ovnlecvr2 39500 hoiqssbl 39515 hoimbl 39521 |
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