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Theorem funeu2 6585
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 5154 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2 funeu 6584 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
3 df-br 5154 . . . 4 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
43eubii 2574 . . 3 (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
52, 4sylib 217 . 2 ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
61, 5sylan2br 593 1 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2099  ∃!weu 2557  cop 4639   class class class wbr 5153  Fun wfun 6548
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 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-fun 6556
This theorem is referenced by:  funssres  6603
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