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Mirrors > Home > MPE Home > Th. List > Mathboxes > f002 | Structured version Visualization version GIF version |
Description: A function with an empty codomain must have empty domain. (Contributed by Zhi Wang, 1-Oct-2024.) |
Ref | Expression |
---|---|
f002.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
f002 | ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f002.1 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | feq3 6576 | . . 3 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅)) | |
3 | f00 6649 | . . . 4 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) | |
4 | 3 | simprbi 497 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅) |
5 | 2, 4 | syl6bi 252 | . 2 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 → 𝐴 = ∅)) |
6 | 1, 5 | syl5com 31 | 1 ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∅c0 4257 ⟶wf 6423 |
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 5222 ax-nul 5229 ax-pr 5351 |
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-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-br 5075 df-opab 5137 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-fun 6429 df-fn 6430 df-f 6431 |
This theorem is referenced by: fullthinc2 46284 thincciso 46286 |
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