Theorem fneu 6657

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 6561	. . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐵𝐹𝑦)
2	1	adantr 482	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃*𝑦 𝐵𝐹𝑦)
3		eldmg 5897	. . . . . 6 ⊢ (𝐵 ∈ dom 𝐹 → (𝐵 ∈ dom 𝐹 ↔ ∃𝑦 𝐵𝐹𝑦))
4	3	ibi 267	. . . . 5 ⊢ (𝐵 ∈ dom 𝐹 → ∃𝑦 𝐵𝐹𝑦)
5	4	adantl 483	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃𝑦 𝐵𝐹𝑦)
6		exmoeub 2575	. . . 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 231	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃!𝑦 𝐵𝐹𝑦)
9	8	funfni 6653	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∃wex 1782   ∈ wcel 2107  ∃*wmo 2533  ∃!weu 2563   class class class wbr 5148  dom cdm 5676  Fun wfun 6535   Fn wfn 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-fun 6543  df-fn 6544

This theorem is referenced by:  fneu2  6658  fnbrfvb  6942  mapsnd  8877  fnbrafv2b  45943