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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 5298 | . . . 4 ⊢ ∅ ∈ V | |
2 | eqeq2 2736 | . . . . . 6 ⊢ (𝑔 = ∅ → (𝑓 = 𝑔 ↔ 𝑓 = ∅)) | |
3 | 2 | imbi2d 340 | . . . . 5 ⊢ (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
4 | 3 | albidv 1915 | . . . 4 ⊢ (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
5 | 1, 4 | spcev 3588 | . . 3 ⊢ (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)) |
6 | f00 6764 | . . . 4 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
7 | 6 | simplbi 497 | . . 3 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅) |
8 | 5, 7 | mpg 1791 | . 2 ⊢ ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) |
9 | df-mo 2526 | . 2 ⊢ (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)) | |
10 | 8, 9 | mpbir 230 | 1 ⊢ ∃*𝑓 𝑓:𝐴⟶∅ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ∃wex 1773 ∃*wmo 2524 ∅c0 4315 ⟶wf 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-fun 6536 df-fn 6537 df-f 6538 |
This theorem is referenced by: mof02 47717 |
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