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| Mirrors > Home > MPE Home > Th. List > Mathboxes > f102g | Structured version Visualization version GIF version | ||
| Description: A function that maps the empty set to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.) |
| Ref | Expression |
|---|---|
| f102g | ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | feq2 6717 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∅⟶𝐵)) | |
| 2 | 1 | biimpa 476 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅⟶𝐵) |
| 3 | f0bi 6791 | . . . 4 ⊢ (𝐹:∅⟶𝐵 ↔ 𝐹 = ∅) | |
| 4 | f10 6881 | . . . . 5 ⊢ ∅:∅–1-1→𝐵 | |
| 5 | f1eq1 6799 | . . . . 5 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐵 ↔ ∅:∅–1-1→𝐵)) | |
| 6 | 4, 5 | mpbiri 258 | . . . 4 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐵) |
| 7 | 3, 6 | sylbi 217 | . . 3 ⊢ (𝐹:∅⟶𝐵 → 𝐹:∅–1-1→𝐵) |
| 8 | 2, 7 | syl 17 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅–1-1→𝐵) |
| 9 | f1eq2 6800 | . . 3 ⊢ (𝐴 = ∅ → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵)) | |
| 10 | 9 | adantr 480 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵)) |
| 11 | 8, 10 | mpbird 257 | 1 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∅c0 4333 ⟶wf 6557 –1-1→wf1 6558 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 |
| This theorem is referenced by: f1mo 48762 |
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