MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funeu Structured version   Visualization version   GIF version

Theorem funeu 6359
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 6351 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 5791 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
31, 2sylan 583 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
4 eldmg 5744 . . . 4 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ dom 𝐹 ↔ ∃𝑦 𝐴𝐹𝑦))
54ibi 270 . . 3 (𝐴 ∈ dom 𝐹 → ∃𝑦 𝐴𝐹𝑦)
63, 5syl 17 . 2 ((Fun 𝐹𝐴𝐹𝐵) → ∃𝑦 𝐴𝐹𝑦)
7 funmo 6350 . . . 4 (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
87adantr 484 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
9 moeu 2667 . . 3 (∃*𝑦 𝐴𝐹𝑦 ↔ (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
108, 9sylib 221 . 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 399  wex 1781  wcel 2114  ∃*wmo 2620  ∃!weu 2652   class class class wbr 5042  dom cdm 5532  Rel wrel 5537  Fun wfun 6328
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-fun 6336
This theorem is referenced by:  funeu2  6360  funbrfv  6698  frege124d  40392  funbrafv2  43742
  Copyright terms: Public domain W3C validator