Theorem funeu2 6075

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 4805	. 2 ⊢ (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2		funeu 6074	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
3		df-br 4805	. . . 4 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
4	3	eubii 2629	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
5	2, 4	sylib 208	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
6	1, 5	sylan2br 494	1 ⊢ ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2139  ∃!weu 2607  ⟨cop 4327   class class class wbr 4804  Fun wfun 6043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-fun 6051

This theorem is referenced by:  funssres  6091