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Mirrors > Home > MPE Home > Th. List > fneu | 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 |
---|---|
fneu | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmo 6563 | . . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐵𝐹𝑦) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃*𝑦 𝐵𝐹𝑦) |
3 | eldmg 5898 | . . . . . 6 ⊢ (𝐵 ∈ dom 𝐹 → (𝐵 ∈ dom 𝐹 ↔ ∃𝑦 𝐵𝐹𝑦)) | |
4 | 3 | ibi 266 | . . . . 5 ⊢ (𝐵 ∈ dom 𝐹 → ∃𝑦 𝐵𝐹𝑦) |
5 | 4 | adantl 482 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃𝑦 𝐵𝐹𝑦) |
6 | exmoeub 2574 | . . . 4 ⊢ (∃𝑦 𝐵𝐹𝑦 → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦)) |
8 | 2, 7 | mpbid 231 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃!𝑦 𝐵𝐹𝑦) |
9 | 8 | funfni 6655 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 ∃*wmo 2532 ∃!weu 2562 class class class wbr 5148 dom cdm 5676 Fun wfun 6537 Fn wfn 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-fun 6545 df-fn 6546 |
This theorem is referenced by: fneu2 6660 fnbrfvb 6944 mapsnd 8879 fnbrafv2b 45946 |
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