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Theorem nelrnfvne 6314
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 6313 . 2 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) ∈ ran 𝐹)
2 elnelne2 2904 . 2 (((𝐹𝑋) ∈ ran 𝐹𝑌 ∉ ran 𝐹) → (𝐹𝑋) ≠ 𝑌)
31, 2stoic3 1698 1 ((Fun 𝐹𝑋 ∈ dom 𝐹𝑌 ∉ ran 𝐹) → (𝐹𝑋) ≠ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036  wcel 1987  wne 2790  wnel 2893  dom cdm 5079  ran crn 5080  Fun wfun 5846  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860
This theorem is referenced by:  fveqdmss  6315  fveqressseq  6316
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