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Theorem elfvunirn 6920
Description: A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6925). (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 4333 . . . 4 (𝐵 ∈ (𝐹𝐴) → (𝐹𝐴) ≠ ∅)
2 fvn0fvelrn 6919 . . . 4 ((𝐹𝐴) ≠ ∅ → (𝐹𝐴) ∈ ran 𝐹)
3 elssuni 4940 . . . 4 ((𝐹𝐴) ∈ ran 𝐹 → (𝐹𝐴) ⊆ ran 𝐹)
41, 2, 33syl 18 . . 3 (𝐵 ∈ (𝐹𝐴) → (𝐹𝐴) ⊆ ran 𝐹)
54sseld 3980 . 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 2941  wss 3947  c0 4321   cuni 4907  ran crn 5676  cfv 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-cnv 5683  df-dm 5685  df-rn 5686  df-iota 6492  df-fv 6548
This theorem is referenced by:  fvssunirn  6921  elrnustOLD  23711  ustbas  23714  utopval  23719  tusval  23752  ucnval  23764  iscfilu  23775  metuval  24040  metidval  32808  pstmval  32813  measbasedom  33138  sxbrsigalem0  33208
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