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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 7158 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) | |
2 | 1 | biimpa 476 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
3 | 2 | simpld 494 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → (𝐹‘𝐴) ∈ 𝐵) |
4 | f1sng 6891 | . . 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 6800 | . . 3 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹:{𝐴}–1-1→𝐵 ↔ {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵)) | |
7 | 2, 6 | simpl2im 503 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → (𝐹:{𝐴}–1-1→𝐵 ↔ {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵)) |
8 | 5, 7 | mpbird 257 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → 𝐹:{𝐴}–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 ⟶wf 6559 –1-1→wf1 6560 ‘cfv 6563 |
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-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-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 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-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: f1mo 48683 |
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