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Theorem elfvunirn 6937
Description: A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6942). (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 4340 . . . 4 (𝐵 ∈ (𝐹𝐴) → (𝐹𝐴) ≠ ∅)
2 fvn0fvelrn 6936 . . . 4 ((𝐹𝐴) ≠ ∅ → (𝐹𝐴) ∈ ran 𝐹)
3 elssuni 4936 . . . 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 2107  wne 2939  wss 3950  c0 4332   cuni 4906  ran crn 5685  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695  df-iota 6513  df-fv 6568
This theorem is referenced by:  fvssunirn  6938  elrnustOLD  24234  ustbas  24237  utopval  24242  tusval  24275  ucnval  24287  iscfilu  24298  metuval  24563  metidval  33890  pstmval  33895  measbasedom  34204  sxbrsigalem0  34274
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