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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 4989 | . . . 4 ⊢ (𝐹:A⟶B → 𝐹 Fn A) | |
2 | fneu2 4947 | . . . 4 ⊢ ((𝐹 Fn A ∧ 𝐶 ∈ A) → ∃!y〈𝐶, y〉 ∈ 𝐹) | |
3 | 1, 2 | sylan 267 | . . 3 ⊢ ((𝐹:A⟶B ∧ 𝐶 ∈ A) → ∃!y〈𝐶, y〉 ∈ 𝐹) |
4 | opelf 5005 | . . . . . . . 8 ⊢ ((𝐹:A⟶B ∧ 〈𝐶, y〉 ∈ 𝐹) → (𝐶 ∈ A ∧ y ∈ B)) | |
5 | 4 | simprd 107 | . . . . . . 7 ⊢ ((𝐹:A⟶B ∧ 〈𝐶, y〉 ∈ 𝐹) → y ∈ B) |
6 | 5 | ex 108 | . . . . . 6 ⊢ (𝐹:A⟶B → (〈𝐶, y〉 ∈ 𝐹 → y ∈ B)) |
7 | 6 | pm4.71rd 374 | . . . . 5 ⊢ (𝐹:A⟶B → (〈𝐶, y〉 ∈ 𝐹 ↔ (y ∈ B ∧ 〈𝐶, y〉 ∈ 𝐹))) |
8 | 7 | eubidv 1905 | . . . 4 ⊢ (𝐹:A⟶B → (∃!y〈𝐶, y〉 ∈ 𝐹 ↔ ∃!y(y ∈ B ∧ 〈𝐶, y〉 ∈ 𝐹))) |
9 | 8 | adantr 261 | . . 3 ⊢ ((𝐹:A⟶B ∧ 𝐶 ∈ A) → (∃!y〈𝐶, y〉 ∈ 𝐹 ↔ ∃!y(y ∈ B ∧ 〈𝐶, y〉 ∈ 𝐹))) |
10 | 3, 9 | mpbid 135 | . 2 ⊢ ((𝐹:A⟶B ∧ 𝐶 ∈ A) → ∃!y(y ∈ B ∧ 〈𝐶, y〉 ∈ 𝐹)) |
11 | df-reu 2307 | . 2 ⊢ (∃!y ∈ B 〈𝐶, y〉 ∈ 𝐹 ↔ ∃!y(y ∈ B ∧ 〈𝐶, y〉 ∈ 𝐹)) | |
12 | 10, 11 | sylibr 137 | 1 ⊢ ((𝐹:A⟶B ∧ 𝐶 ∈ A) → ∃!y ∈ B 〈𝐶, y〉 ∈ 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1390 ∃!weu 1897 ∃!wreu 2302 〈cop 3370 Fn wfn 4840 ⟶wf 4841 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-f 4849 |
This theorem is referenced by: fsn 5278 f1ofveu 5443 |
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