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Theorem funeu2 6579
Description: There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
funeu2 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem funeu2
StepHypRef Expression
1 df-br 5149 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2 funeu 6578 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
3 df-br 5149 . . . 4 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
43eubii 2575 . . 3 (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
52, 4sylib 217 . 2 ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
61, 5sylan2br 594 1 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  ∃!weu 2558  cop 4635   class class class wbr 5148  Fun wfun 6542
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 5299  ax-nul 5306  ax-pr 5429
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 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-fun 6550
This theorem is referenced by:  funssres  6597
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