MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nelrnfvne Structured version   Visualization version   GIF version

Theorem nelrnfvne 7097
Description: A function value cannot be any element not contained in the range of the function. (Contributed by AV, 28-Jan-2020.)
Assertion
Ref Expression
nelrnfvne ((Fun 𝐹𝑋 ∈ dom 𝐹𝑌 ∉ ran 𝐹) → (𝐹𝑋) ≠ 𝑌)

Proof of Theorem nelrnfvne
StepHypRef Expression
1 fvelrn 7096 . 2 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) ∈ ran 𝐹)
2 elnelne2 3056 . 2 (((𝐹𝑋) ∈ ran 𝐹𝑌 ∉ ran 𝐹) → (𝐹𝑋) ≠ 𝑌)
31, 2stoic3 1773 1 ((Fun 𝐹𝑋 ∈ dom 𝐹𝑌 ∉ ran 𝐹) → (𝐹𝑋) ≠ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2106  wne 2938  wnel 3044  dom cdm 5689  ran crn 5690  Fun wfun 6557  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  fveqdmss  7098  fveqressseq  7099
  Copyright terms: Public domain W3C validator