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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 6669 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fneu2 6614 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹) | |
3 | 1, 2 | sylan 581 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹) |
4 | opelf 6704 | . . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
5 | 4 | simprd 497 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ 𝐵) |
6 | 5 | ex 414 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ 𝐵)) |
7 | 6 | pm4.71rd 564 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))) |
8 | 7 | eubidv 2585 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))) |
9 | 8 | adantr 482 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))) |
10 | 3, 9 | mpbid 231 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)) |
11 | df-reu 3355 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)) | |
12 | 10, 11 | sylibr 233 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∃!weu 2567 ∃!wreu 3352 ⟨cop 4593 Fn wfn 6492 ⟶wf 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 |
This theorem is referenced by: fsn 7082 f1ofveu 7352 |
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