Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6374 | . . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐵𝐹𝑦) | |
2 | 1 | adantr 484 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃*𝑦 𝐵𝐹𝑦) |
3 | eldmg 5752 | . . . . . 6 ⊢ (𝐵 ∈ dom 𝐹 → (𝐵 ∈ dom 𝐹 ↔ ∃𝑦 𝐵𝐹𝑦)) | |
4 | 3 | ibi 270 | . . . . 5 ⊢ (𝐵 ∈ dom 𝐹 → ∃𝑦 𝐵𝐹𝑦) |
5 | 4 | adantl 485 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃𝑦 𝐵𝐹𝑦) |
6 | exmoeub 2579 | . . . 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 235 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃!𝑦 𝐵𝐹𝑦) |
9 | 8 | funfni 6462 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∃wex 1787 ∈ wcel 2112 ∃*wmo 2537 ∃!weu 2567 class class class wbr 5039 dom cdm 5536 Fun wfun 6352 Fn wfn 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-fun 6360 df-fn 6361 |
This theorem is referenced by: fneu2 6467 fnbrfvb 6743 mapsnd 8545 fnbrafv2b 44355 |
Copyright terms: Public domain | W3C validator |