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Mirrors > Home > ILE Home > Th. List > feu | 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 5322 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fneu2 5278 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹) | |
3 | 1, 2 | sylan 281 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹) |
4 | opelf 5344 | . . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
5 | 4 | simprd 113 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹) → 𝑦 ∈ 𝐵) |
6 | 5 | ex 114 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝑦〉 ∈ 𝐹 → 𝑦 ∈ 𝐵)) |
7 | 6 | pm4.71rd 392 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝑦〉 ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
8 | 7 | eubidv 2014 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
9 | 8 | adantr 274 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (∃!𝑦〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹))) |
10 | 3, 9 | mpbid 146 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹)) |
11 | df-reu 2442 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 〈𝐶, 𝑦〉 ∈ 𝐹)) | |
12 | 10, 11 | sylibr 133 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∃!weu 2006 ∈ wcel 2128 ∃!wreu 2437 〈cop 3564 Fn wfn 5168 ⟶wf 5169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-fun 5175 df-fn 5176 df-f 5177 |
This theorem is referenced by: fsn 5642 f1ofveu 5815 |
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