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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 6506 | . . 3 ⊢ (𝐵 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∅)) | |
3 | f00 6579 | . . . 4 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) | |
4 | 3 | simprbi 500 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅) |
5 | 2, 4 | syl6bi 256 | . 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 1543 ∅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: fullthinc2 45944 thincciso 45946 |
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