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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 5943 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | releldm 5390 | . . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹) | |
3 | 1, 2 | sylan 487 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹) |
4 | eldmg 5351 | . . . 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 5942 | . . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | |
8 | 7 | adantr 480 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦) |
9 | df-mo 2503 | . . 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 383 ∃wex 1744 ∈ wcel 2030 ∃!weu 2498 ∃*wmo 2499 class class class wbr 4685 dom cdm 5143 Rel wrel 5148 Fun wfun 5920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-fun 5928 |
This theorem is referenced by: funeu2 5952 funbrfv 6272 frege124d 38370 |
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