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Theorem fneu 6659
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 6563 . . . 4 (Fun 𝐹 → ∃*𝑦 𝐵𝐹𝑦)
21adantr 481 . . 3 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃*𝑦 𝐵𝐹𝑦)
3 eldmg 5898 . . . . . 6 (𝐵 ∈ dom 𝐹 → (𝐵 ∈ dom 𝐹 ↔ ∃𝑦 𝐵𝐹𝑦))
43ibi 266 . . . . 5 (𝐵 ∈ dom 𝐹 → ∃𝑦 𝐵𝐹𝑦)
54adantl 482 . . . 4 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃𝑦 𝐵𝐹𝑦)
6 exmoeub 2574 . . . 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 6655 1 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!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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