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Theorem funeu 5814
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 5807 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 5266 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
31, 2sylan 486 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
4 eldmg 5228 . . . 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 5806 . . . 4 (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
87adantr 479 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
9 df-mo 2462 . . 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 1976  ∃!weu 2457  ∃*wmo 2458   class class class wbr 4577  dom cdm 5028  Rel wrel 5033  Fun wfun 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
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 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-fun 5792
This theorem is referenced by:  funeu2  5815  funbrfv  6129  frege124d  36868
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