Theorem elfvunirn 6946

Description: A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6950). (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 4290	. . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ≠ ∅)
2		fvn0fvelrn 6945	. . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹‘𝐴) ∈ ran 𝐹)
3		elssuni 4904	. . . 4 ⊢ ((𝐹‘𝐴) ∈ ran 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
4	1, 2, 3	3syl 19	. . 3 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
5	4	sseld 3912	. 2 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹))
6	5	pm2.43i 53	1 ⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900  ∅c0 4281  ∪ cuni 4872  ran crn 5659  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-cnv 5666  df-dm 5668  df-rn 5669  df-iota 6520  df-fv 6574

This theorem is used by:  fvssunirn  6947  ustbas  24601  utopval  24606  tusval  24639  ucnval  24650  iscfilu  24661  metuval  24923  metidval  34633  pstmval  34638  measbasedom  34946  sxbrsigalem0  35014  tmachlem-exlargecover  48227