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Theorem nelrnfvne 6839
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 6838 . 2 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) ∈ ran 𝐹)
2 elnelne2 3134 . 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 1083  wcel 2110  wne 3016  wnel 3123  dom cdm 5549  ran crn 5550  Fun wfun 6343  cfv 6349
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357
This theorem is referenced by:  fveqdmss  6840  fveqressseq  6841
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