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Theorem fneu 6291
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 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐵
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fneu
StepHypRef Expression
1 funmo 6201 . . . 4 (Fun 𝐹 → ∃*𝑦 𝐵𝐹𝑦)
21adantr 473 . . 3 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃*𝑦 𝐵𝐹𝑦)
3 eldmg 5613 . . . . . 6 (𝐵 ∈ dom 𝐹 → (𝐵 ∈ dom 𝐹 ↔ ∃𝑦 𝐵𝐹𝑦))
43ibi 259 . . . . 5 (𝐵 ∈ dom 𝐹 → ∃𝑦 𝐵𝐹𝑦)
54adantl 474 . . . 4 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃𝑦 𝐵𝐹𝑦)
6 exmoeub 2600 . . . 4 (∃𝑦 𝐵𝐹𝑦 → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦))
75, 6syl 17 . . 3 ((Fun 𝐹𝐵 ∈ dom 𝐹) → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦))
82, 7mpbid 224 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃!𝑦 𝐵𝐹𝑦)
98funfni 6287 1 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wex 1743  wcel 2051  ∃*wmo 2546  ∃!weu 2584   class class class wbr 4925  dom cdm 5403  Fun wfun 6179   Fn wfn 6180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-fun 6187  df-fn 6188
This theorem is referenced by:  fneu2  6292  fnbrfvb  6545  mapsnd  8246  fnimasnd  38604  fnbrafv2b  42887
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