Theorem fneu 6227

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 6138	. . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐵𝐹𝑦)
2	1	adantr 474	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃*𝑦 𝐵𝐹𝑦)
3		eldmg 5550	. . . . . 6 ⊢ (𝐵 ∈ dom 𝐹 → (𝐵 ∈ dom 𝐹 ↔ ∃𝑦 𝐵𝐹𝑦))
4	3	ibi 259	. . . . 5 ⊢ (𝐵 ∈ dom 𝐹 → ∃𝑦 𝐵𝐹𝑦)
5	4	adantl 475	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃𝑦 𝐵𝐹𝑦)
6		exmoeub 2652	. . . 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 224	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃!𝑦 𝐵𝐹𝑦)
9	8	funfni 6223	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∃wex 1880   ∈ wcel 2166  ∃*wmo 2602  ∃!weu 2638   class class class wbr 4872  dom cdm 5341  Fun wfun 6116   Fn wfn 6117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-fun 6124  df-fn 6125

This theorem is referenced by:  fneu2  6228  fnbrfvb  6481  mapsnd  8163  fnbrafv2b  42149