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Mirrors > Home > MPE Home > Th. List > Mathboxes > mof0 | Structured version Visualization version GIF version |
Description: There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
mof0 | ⊢ ∃*𝑓 𝑓:𝐴⟶∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5309 | . . . 4 ⊢ ∅ ∈ V | |
2 | eqeq2 2745 | . . . . . 6 ⊢ (𝑔 = ∅ → (𝑓 = 𝑔 ↔ 𝑓 = ∅)) | |
3 | 2 | imbi2d 340 | . . . . 5 ⊢ (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
4 | 3 | albidv 1916 | . . . 4 ⊢ (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
5 | 1, 4 | spcev 3606 | . . 3 ⊢ (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)) |
6 | f00 6786 | . . . 4 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
7 | 6 | simplbi 497 | . . 3 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅) |
8 | 5, 7 | mpg 1792 | . 2 ⊢ ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) |
9 | df-mo 2536 | . 2 ⊢ (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)) | |
10 | 8, 9 | mpbir 231 | 1 ⊢ ∃*𝑓 𝑓:𝐴⟶∅ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1533 = wceq 1535 ∃wex 1774 ∃*wmo 2534 ∅c0 4339 ⟶wf 6555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5151 df-opab 5213 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6561 df-fn 6562 df-f 6563 |
This theorem is referenced by: mof02 48597 |
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