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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 6931 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}))) | |
2 | 1 | biimpa 480 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
3 | 2 | simpld 498 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → (𝐹‘𝐴) ∈ 𝐵) |
4 | f1sng 6680 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ 𝐵) → {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵) | |
5 | 3, 4 | syldan 594 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵) |
6 | f1eq1 6588 | . . 3 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹:{𝐴}–1-1→𝐵 ↔ {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵)) | |
7 | 2, 6 | simpl2im 507 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → (𝐹:{𝐴}–1-1→𝐵 ↔ {〈𝐴, (𝐹‘𝐴)〉}:{𝐴}–1-1→𝐵)) |
8 | 5, 7 | mpbird 260 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → 𝐹:{𝐴}–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {csn 4527 〈cop 4533 ⟶wf 6354 –1-1→wf1 6355 ‘cfv 6358 |
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-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 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-uni 4806 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-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 |
This theorem is referenced by: f1mo 45796 |
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