Theorem elfvunirn 6847

Description: A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6851). (Contributed by Thierry Arnoux, 13-Nov-2016.) Remove functionhood antecedent. (Revised by SN, 10-Jan-2025.)

Assertion

Ref	Expression
elfvunirn	⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹)

Proof of Theorem elfvunirn

Step	Hyp	Ref	Expression
1		ne0i 4286	. . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ≠ ∅)
2		fvn0fvelrn 6846	. . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹‘𝐴) ∈ ran 𝐹)
3		elssuni 4884	. . . 4 ⊢ ((𝐹‘𝐴) ∈ ran 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
5	4	sseld 3928	. 2 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹))
6	5	pm2.43i 52	1 ⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹)

Syntax hints:   → wi 4   ∈ wcel 2111   ≠ wne 2928   ⊆ wss 3897  ∅c0 4278  ∪ cuni 4854  ran crn 5612  ‘cfv 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-cnv 5619  df-dm 5621  df-rn 5622  df-iota 6432  df-fv 6484

This theorem is referenced by:  fvssunirn  6848  ustbas  24137  utopval  24142  tusval  24175  ucnval  24186  iscfilu  24197  metuval  24459  metidval  33895  pstmval  33900  measbasedom  34207  sxbrsigalem0  34276