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Theorem fnfvelrnd 42790
Description: A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
fnfvelrnd.1 (𝜑𝐹 Fn 𝐴)
fnfvelrnd.2 (𝜑𝐵𝐴)
Assertion
Ref Expression
fnfvelrnd (𝜑 → (𝐹𝐵) ∈ ran 𝐹)

Proof of Theorem fnfvelrnd
StepHypRef Expression
1 fnfvelrnd.1 . 2 (𝜑𝐹 Fn 𝐴)
2 fnfvelrnd.2 . 2 (𝜑𝐵𝐴)
3 fnfvelrn 6955 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) ∈ ran 𝐹)
41, 2, 3syl2anc 584 1 (𝜑 → (𝐹𝐵) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  ran crn 5591   Fn wfn 6427  cfv 6432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-iota 6390  df-fun 6434  df-fn 6435  df-fv 6440
This theorem is referenced by:  limsupgtlem  43300
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