Theorem fneu 6602

Description: There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fneu	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦)

Distinct variable groups:   𝑦,𝐹   𝑦,𝐵

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fneu

Step	Hyp	Ref	Expression
1		funmo 6508	. . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐵𝐹𝑦)
2	1	adantr 480	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃*𝑦 𝐵𝐹𝑦)
3		eldmg 5847	. . . . . 6 ⊢ (𝐵 ∈ dom 𝐹 → (𝐵 ∈ dom 𝐹 ↔ ∃𝑦 𝐵𝐹𝑦))
4	3	ibi 267	. . . . 5 ⊢ (𝐵 ∈ dom 𝐹 → ∃𝑦 𝐵𝐹𝑦)
5	4	adantl 481	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃𝑦 𝐵𝐹𝑦)
6		exmoeub 2580	. . . 4 ⊢ (∃𝑦 𝐵𝐹𝑦 → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦))
7	5, 6	syl 17	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦))
8	2, 7	mpbid 232	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃!𝑦 𝐵𝐹𝑦)
9	8	funfni 6598	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2537  ∃!weu 2568   class class class wbr 5098  dom cdm 5624  Fun wfun 6486   Fn wfn 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-fun 6494  df-fn 6495

This theorem is referenced by:  fneu2  6603  fnbrfvb  6884  mapsnd  8824  brpermmodel  45244  fnbrafv2b  47494