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Theorem funeu 6564
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
funeu ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem funeu
StepHypRef Expression
1 funrel 6556 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 5934 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
31, 2sylan 579 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
4 eldmg 5889 . . . 4 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ dom 𝐹 ↔ ∃𝑦 𝐴𝐹𝑦))
54ibi 267 . . 3 (𝐴 ∈ dom 𝐹 → ∃𝑦 𝐴𝐹𝑦)
63, 5syl 17 . 2 ((Fun 𝐹𝐴𝐹𝐵) → ∃𝑦 𝐴𝐹𝑦)
7 funmo 6554 . . . 4 (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
87adantr 480 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
9 moeu 2569 . . 3 (∃*𝑦 𝐴𝐹𝑦 ↔ (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
108, 9sylib 217 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
116, 10mpd 15 1 ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1773  wcel 2098  ∃*wmo 2524  ∃!weu 2554   class class class wbr 5139  dom cdm 5667  Rel wrel 5672  Fun wfun 6528
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
This theorem is referenced by:  funeu2  6565  funbrfv  6933  frege124d  43061  funbrafv2  46500
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