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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 5305 | . . . 4 ⊢ ∅ ∈ V | |
| 2 | eqeq2 2748 | . . . . . 6 ⊢ (𝑔 = ∅ → (𝑓 = 𝑔 ↔ 𝑓 = ∅)) | |
| 3 | 2 | imbi2d 340 | . . . . 5 ⊢ (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
| 4 | 3 | albidv 1920 | . . . 4 ⊢ (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
| 5 | 1, 4 | spcev 3605 | . . 3 ⊢ (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)) |
| 6 | f00 6788 | . . . 4 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
| 7 | 6 | simplbi 497 | . . 3 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅) |
| 8 | 5, 7 | mpg 1797 | . 2 ⊢ ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) |
| 9 | df-mo 2539 | . 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 1538 = wceq 1540 ∃wex 1779 ∃*wmo 2537 ∅c0 4332 ⟶wf 6555 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5142 df-opab 5204 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-fun 6561 df-fn 6562 df-f 6563 |
| This theorem is referenced by: mof02 48721 |
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