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Theorem funeu2 6153
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 4876 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2 funeu 6152 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
3 df-br 4876 . . . 4 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
43eubii 2658 . . 3 (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
52, 4sylib 210 . 2 ((Fun 𝐹𝐴𝐹𝐵) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
61, 5sylan2br 588 1 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2164  ∃!weu 2639  cop 4405   class class class wbr 4875  Fun wfun 6121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-fun 6129
This theorem is referenced by:  funssres  6170
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