Theorem fneu2 2729

Description: There is exactly one value of a function.

Assertion

Ref	Expression
fneu2	⊢ ((F Fn A ∧ B ∈ A) → ∃!y⟨B, y⟩ ∈ F)

Distinct variable group(s):   y,F   y,B

Proof of Theorem fneu2

Step	Hyp	Ref	Expression
1		fneu 2728	. 2 ⊢ ((F Fn A ∧ B ∈ A) → ∃!y BFy)
2		df-br 2063	. . 3 ⊢ (BFy ↔ ⟨B, y⟩ ∈ F)
3	2	bieu 1014	. 2 ⊢ (∃!y BFy ↔ ∃!y⟨B, y⟩ ∈ F)
4	1, 3	sylib 173	1 ⊢ ((F Fn A ∧ B ∈ A) → ∃!y⟨B, y⟩ ∈ F)

Syntax hints:   → wi 2   ∧ wa 196  ∃!weu 1007   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054   Fn wfn 2417

This theorem is referenced by:  fnopabg 2745  feu 2767  mapsn 3269

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-cnv 2426  df-co 2427  df-dm 2428  df-fun 2432  df-fn 2433

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