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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 6487 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fneu2 6433 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹) | |
3 | 1, 2 | sylan 583 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹) |
4 | opelf 6513 | . . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
5 | 4 | simprd 499 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹) → 𝑦 ∈ 𝐵) |
6 | 5 | ex 416 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝑦〉 ∈ 𝐹 → 𝑦 ∈ 𝐵)) |
7 | 6 | pm4.71rd 566 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝑦〉 ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
8 | 7 | eubidv 2647 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
9 | 8 | adantr 484 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
10 | 3, 9 | mpbid 235 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹)) |
11 | df-reu 3113 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹)) | |
12 | 10, 11 | sylibr 237 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∃!weu 2628 ∃!wreu 3108 〈cop 4531 Fn wfn 6319 ⟶wf 6320 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-reu 3113 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 |
This theorem is referenced by: fsn 6874 f1ofveu 7130 |
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