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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 5185 | . . . 4 ⊢ ∅ ∈ V | |
2 | eqeq2 2748 | . . . . . 6 ⊢ (𝑔 = ∅ → (𝑓 = 𝑔 ↔ 𝑓 = ∅)) | |
3 | 2 | imbi2d 344 | . . . . 5 ⊢ (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
4 | 3 | albidv 1928 | . . . 4 ⊢ (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅))) |
5 | 1, 4 | spcev 3511 | . . 3 ⊢ (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)) |
6 | f00 6579 | . . . 4 ⊢ (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
7 | 6 | simplbi 501 | . . 3 ⊢ (𝑓:𝐴⟶∅ → 𝑓 = ∅) |
8 | 5, 7 | mpg 1805 | . 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 234 | 1 ⊢ ∃*𝑓 𝑓:𝐴⟶∅ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 = wceq 1543 ∃wex 1787 ∃*wmo 2537 ∅c0 4223 ⟶wf 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-fun 6360 df-fn 6361 df-f 6362 |
This theorem is referenced by: mof02 45782 |
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