Theorem funeu2 6572

Description: There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)

Assertion

Ref	Expression
funeu2	⊢ ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem funeu2

Step	Hyp	Ref	Expression
1		df-br 5124	. 2 ⊢ (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2		funeu 6571	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
3		df-br 5124	. . . 4 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
4	3	eubii 2583	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
5	2, 4	sylib 218	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
6	1, 5	sylan2br 595	1 ⊢ ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107  ∃!weu 2566  ⟨cop 4612   class class class wbr 5123  Fun wfun 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-fun 6543

This theorem is referenced by:  funssres  6590