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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 3058 . 2 (((𝐹𝑋) ∈ ran 𝐹𝑌 ∉ ran 𝐹) → (𝐹𝑋) ≠ 𝑌)
31, 2stoic3 1776 1 ((Fun 𝐹𝑋 ∈ dom 𝐹𝑌 ∉ ran 𝐹) → (𝐹𝑋) ≠ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2108  wne 2940  wnel 3046  dom cdm 5685  ran crn 5686  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  fveqdmss  7098  fveqressseq  7099
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