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Theorem fneu 6650
Description: There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fneu ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐵
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fneu
StepHypRef Expression
1 funmo 6554 . . . 4 (Fun 𝐹 → ∃*𝑦 𝐵𝐹𝑦)
21adantr 480 . . 3 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃*𝑦 𝐵𝐹𝑦)
3 eldmg 5889 . . . . . 6 (𝐵 ∈ dom 𝐹 → (𝐵 ∈ dom 𝐹 ↔ ∃𝑦 𝐵𝐹𝑦))
43ibi 267 . . . . 5 (𝐵 ∈ dom 𝐹 → ∃𝑦 𝐵𝐹𝑦)
54adantl 481 . . . 4 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃𝑦 𝐵𝐹𝑦)
6 exmoeub 2566 . . . 4 (∃𝑦 𝐵𝐹𝑦 → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦))
75, 6syl 17 . . 3 ((Fun 𝐹𝐵 ∈ dom 𝐹) → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦))
82, 7mpbid 231 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃!𝑦 𝐵𝐹𝑦)
98funfni 6646 1 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wex 1773  wcel 2098  ∃*wmo 2524  ∃!weu 2554   class class class wbr 5139  dom cdm 5667  Fun wfun 6528   Fn wfn 6529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-fun 6536  df-fn 6537
This theorem is referenced by:  fneu2  6651  fnbrfvb  6935  mapsnd  8877  fnbrafv2b  46466
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