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Mirrors > Home > MPE Home > Th. List > feu | Structured version Visualization version GIF version |
Description: There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.) |
Ref | Expression |
---|---|
feu | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6737 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fneu2 6680 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹) | |
3 | 1, 2 | sylan 580 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹) |
4 | opelf 6770 | . . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
5 | 4 | simprd 495 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹) → 𝑦 ∈ 𝐵) |
6 | 5 | ex 412 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝑦〉 ∈ 𝐹 → 𝑦 ∈ 𝐵)) |
7 | 6 | pm4.71rd 562 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝑦〉 ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
8 | 7 | eubidv 2584 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
10 | 3, 9 | mpbid 232 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹)) |
11 | df-reu 3379 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹)) | |
12 | 10, 11 | sylibr 234 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∃!weu 2566 ∃!wreu 3376 〈cop 4637 Fn wfn 6558 ⟶wf 6559 |
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-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-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-reu 3379 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 |
This theorem is referenced by: fdmeu 6965 fsn 7155 f1ofveu 7425 |
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