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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 6718 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∅⟶𝐵)) | |
2 | 1 | biimpa 476 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅⟶𝐵) |
3 | f0bi 6792 | . . . 4 ⊢ (𝐹:∅⟶𝐵 ↔ 𝐹 = ∅) | |
4 | f10 6882 | . . . . 5 ⊢ ∅:∅–1-1→𝐵 | |
5 | f1eq1 6800 | . . . . 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 6801 | . . 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 1537 ∅c0 4339 ⟶wf 6559 –1-1→wf1 6560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 |
This theorem is referenced by: f1mo 48683 |
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