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Theorem elfvunirn 6923
Description: A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6928). (Contributed by Thierry Arnoux, 13-Nov-2016.) Remove functionhood antecedent. (Revised by SN, 10-Jan-2025.)
Assertion
Ref Expression
elfvunirn (𝐵 ∈ (𝐹𝐴) → 𝐵 ran 𝐹)

Proof of Theorem elfvunirn
StepHypRef Expression
1 ne0i 4334 . . . 4 (𝐵 ∈ (𝐹𝐴) → (𝐹𝐴) ≠ ∅)
2 fvn0fvelrn 6922 . . . 4 ((𝐹𝐴) ≠ ∅ → (𝐹𝐴) ∈ ran 𝐹)
3 elssuni 4941 . . . 4 ((𝐹𝐴) ∈ ran 𝐹 → (𝐹𝐴) ⊆ ran 𝐹)
41, 2, 33syl 18 . . 3 (𝐵 ∈ (𝐹𝐴) → (𝐹𝐴) ⊆ ran 𝐹)
54sseld 3981 . 2 (𝐵 ∈ (𝐹𝐴) → (𝐵 ∈ (𝐹𝐴) → 𝐵 ran 𝐹))
65pm2.43i 52 1 (𝐵 ∈ (𝐹𝐴) → 𝐵 ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2940  wss 3948  c0 4322   cuni 4908  ran crn 5677  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6495  df-fv 6551
This theorem is referenced by:  fvssunirn  6924  elrnustOLD  23728  ustbas  23731  utopval  23736  tusval  23769  ucnval  23781  iscfilu  23792  metuval  24057  metidval  32865  pstmval  32870  measbasedom  33195  sxbrsigalem0  33265
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