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Theorem funeu 5718
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 5711 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 5170 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
31, 2sylan 486 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
4 eldmg 5132 . . . 4 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ dom 𝐹 ↔ ∃𝑦 𝐴𝐹𝑦))
54ibi 254 . . 3 (𝐴 ∈ dom 𝐹 → ∃𝑦 𝐴𝐹𝑦)
63, 5syl 17 . 2 ((Fun 𝐹𝐴𝐹𝐵) → ∃𝑦 𝐴𝐹𝑦)
7 funmo 5710 . . . 4 (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
87adantr 479 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
9 df-mo 2367 . . 3 (∃*𝑦 𝐴𝐹𝑦 ↔ (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
108, 9sylib 206 . 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 382  wex 1694  wcel 1938  ∃!weu 2362  ∃*wmo 2363   class class class wbr 4481  dom cdm 4932  Rel wrel 4937  Fun wfun 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-fun 5696
This theorem is referenced by:  funeu2  5719  funbrfv  6033  frege124d  36973
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