Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdm0OLD | 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 modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdm0OLD | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4790 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | elmapg 7870 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴)) | |
3 | 1, 2 | mpan2 707 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴)) |
4 | 3 | biimpa 501 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (𝐴 ↑𝑚 ∅)) → 𝑓:∅⟶𝐴) |
5 | f0bi 6088 | . . . 4 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 = ∅) | |
6 | 4, 5 | sylib 208 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (𝐴 ↑𝑚 ∅)) → 𝑓 = ∅) |
7 | 6 | ralrimiva 2966 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑓 ∈ (𝐴 ↑𝑚 ∅)𝑓 = ∅) |
8 | f0 6086 | . . . . . 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 7870 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (∅ ∈ (𝐴 ↑𝑚 ∅) ↔ ∅:∅⟶𝐴)) | |
13 | 10, 11, 12 | syl2anc 693 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∅ ∈ (𝐴 ↑𝑚 ∅) ↔ ∅:∅⟶𝐴)) |
14 | 9, 13 | mpbird 247 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ (𝐴 ↑𝑚 ∅)) |
15 | ne0i 3921 | . . . 4 ⊢ (∅ ∈ (𝐴 ↑𝑚 ∅) → (𝐴 ↑𝑚 ∅) ≠ ∅) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) ≠ ∅) |
17 | eqsn 4361 | . . 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 247 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∅c0 3915 {csn 4177 ⟶wf 5884 (class class class)co 6650 ↑𝑚 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 |
This theorem is referenced by: (None) |
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