![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funeu | Structured version Visualization version 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 6049 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | releldm 5497 | . . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹) | |
3 | 1, 2 | sylan 563 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹) |
4 | eldmg 5458 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (𝐴 ∈ dom 𝐹 ↔ ∃𝑦 𝐴𝐹𝑦)) | |
5 | 4 | ibi 256 | . . 3 ⊢ (𝐴 ∈ dom 𝐹 → ∃𝑦 𝐴𝐹𝑦) |
6 | 3, 5 | syl 17 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃𝑦 𝐴𝐹𝑦) |
7 | funmo 6048 | . . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | |
8 | 7 | adantr 466 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦) |
9 | df-mo 2623 | . . 3 ⊢ (∃*𝑦 𝐴𝐹𝑦 ↔ (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦)) | |
10 | 8, 9 | sylib 208 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦)) |
11 | 6, 10 | mpd 15 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∃wex 1852 ∈ wcel 2145 ∃!weu 2618 ∃*wmo 2619 class class class wbr 4787 dom cdm 5250 Rel wrel 5255 Fun wfun 6026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-br 4788 df-opab 4848 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-fun 6034 |
This theorem is referenced by: funeu2 6058 funbrfv 6376 frege124d 38580 |
Copyright terms: Public domain | W3C validator |