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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 5231 | . . . 4 ⊢ ∅ ∈ V | |
2 | eqeq2 2750 | . . . . . 6 ⊢ (𝑔 = ∅ → (𝑓 = 𝑔 ↔ 𝑓 = ∅)) | |
3 | 2 | imbi2d 341 | . . . . 5 ⊢ (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
4 | 3 | albidv 1923 | . . . 4 ⊢ (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
5 | 1, 4 | spcev 3545 | . . 3 ⊢ (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)) |
6 | f00 6656 | . . . 4 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
7 | 6 | simplbi 498 | . . 3 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅) |
8 | 5, 7 | mpg 1800 | . 2 ⊢ ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) |
9 | df-mo 2540 | . 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 1537 = wceq 1539 ∃wex 1782 ∃*wmo 2538 ∅c0 4256 ⟶wf 6429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-fun 6435 df-fn 6436 df-f 6437 |
This theorem is referenced by: mof02 46166 |
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