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Theorem funeu2 6524
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 5104 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2 funeu 6523 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
3 df-br 5104 . . . 4 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
43eubii 2584 . . 3 (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
52, 4sylib 217 . 2 ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
61, 5sylan2br 595 1 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  ∃!weu 2567  cop 4590   class class class wbr 5103  Fun wfun 6487
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-fun 6495
This theorem is referenced by:  funssres  6542
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