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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 6505 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹:𝐴⟶𝐵 ↔ 𝐹:∅⟶𝐵)) | |
2 | 1 | biimpa 480 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅⟶𝐵) |
3 | f0bi 6580 | . . . 4 ⊢ (𝐹:∅⟶𝐵 ↔ 𝐹 = ∅) | |
4 | f10 6671 | . . . . 5 ⊢ ∅:∅–1-1→𝐵 | |
5 | f1eq1 6588 | . . . . 5 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐵 ↔ ∅:∅–1-1→𝐵)) | |
6 | 4, 5 | mpbiri 261 | . . . 4 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐵) |
7 | 3, 6 | sylbi 220 | . . 3 ⊢ (𝐹:∅⟶𝐵 → 𝐹:∅–1-1→𝐵) |
8 | 2, 7 | syl 17 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:∅–1-1→𝐵) |
9 | f1eq2 6589 | . . 3 ⊢ (𝐴 = ∅ → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵)) | |
10 | 9 | adantr 484 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:∅–1-1→𝐵)) |
11 | 8, 10 | mpbird 260 | 1 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∅c0 4223 ⟶wf 6354 –1-1→wf1 6355 |
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 df-f1 6363 |
This theorem is referenced by: f1mo 45796 |
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