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Theorem funeu 6572
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 6564 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 5940 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
31, 2sylan 579 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
4 eldmg 5895 . . . 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 6562 . . . 4 (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
87adantr 480 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
9 moeu 2573 . . 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 1774  wcel 2099  ∃*wmo 2528  ∃!weu 2558   class class class wbr 5142  dom cdm 5672  Rel wrel 5677  Fun wfun 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-fun 6544
This theorem is referenced by:  funeu2  6573  funbrfv  6942  frege124d  43185  funbrafv2  46621
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