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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1sn2g | Structured version Visualization version GIF version |
Description: A function that maps a singleton to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.) |
Ref | Expression |
---|---|
f1sn2g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → 𝐹:{𝐴}–1-1→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsn2g 7080 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) | |
2 | 1 | biimpa 477 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
3 | 2 | simpld 495 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → (𝐹‘𝐴) ∈ 𝐵) |
4 | f1sng 6823 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ 𝐵) → {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵) | |
5 | 3, 4 | syldan 591 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵) |
6 | f1eq1 6730 | . . 3 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹:{𝐴}–1-1→𝐵 ↔ {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵)) | |
7 | 2, 6 | simpl2im 504 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → (𝐹:{𝐴}–1-1→𝐵 ↔ {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵)) |
8 | 5, 7 | mpbird 256 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → 𝐹:{𝐴}–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4584 〈cop 4590 ⟶wf 6489 –1-1→wf1 6490 ‘cfv 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 |
This theorem is referenced by: f1mo 46851 |
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