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Mirrors > Home > ILE Home > Th. List > funeu | GIF version |
Description: There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
funeu | ⊢ ((Fun 𝐹 ∧ A𝐹B) → ∃!y A𝐹y) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 4862 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | releldm 4512 | . . . 4 ⊢ ((Rel 𝐹 ∧ A𝐹B) → A ∈ dom 𝐹) | |
3 | 1, 2 | sylan 267 | . . 3 ⊢ ((Fun 𝐹 ∧ A𝐹B) → A ∈ dom 𝐹) |
4 | eldmg 4473 | . . . 4 ⊢ (A ∈ dom 𝐹 → (A ∈ dom 𝐹 ↔ ∃y A𝐹y)) | |
5 | 4 | ibi 165 | . . 3 ⊢ (A ∈ dom 𝐹 → ∃y A𝐹y) |
6 | 3, 5 | syl 14 | . 2 ⊢ ((Fun 𝐹 ∧ A𝐹B) → ∃y A𝐹y) |
7 | funmo 4860 | . . . 4 ⊢ (Fun 𝐹 → ∃*y A𝐹y) | |
8 | 7 | adantr 261 | . . 3 ⊢ ((Fun 𝐹 ∧ A𝐹B) → ∃*y A𝐹y) |
9 | df-mo 1901 | . . 3 ⊢ (∃*y A𝐹y ↔ (∃y A𝐹y → ∃!y A𝐹y)) | |
10 | 8, 9 | sylib 127 | . 2 ⊢ ((Fun 𝐹 ∧ A𝐹B) → (∃y A𝐹y → ∃!y A𝐹y)) |
11 | 6, 10 | mpd 13 | 1 ⊢ ((Fun 𝐹 ∧ A𝐹B) → ∃!y A𝐹y) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃wex 1378 ∈ wcel 1390 ∃!weu 1897 ∃*wmo 1898 class class class wbr 3755 dom cdm 4288 Rel wrel 4293 Fun wfun 4839 |
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-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-fun 4847 |
This theorem is referenced by: funeu2 4870 funbrfv 5155 |
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