Theorem elfvunirn 6952

Description: A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6957). (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 4364	. . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ≠ ∅)
2		fvn0fvelrn 6951	. . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹‘𝐴) ∈ ran 𝐹)
3		elssuni 4961	. . . 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 4007	. 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 2108   ≠ wne 2946   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ran crn 5701  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-cnv 5708  df-dm 5710  df-rn 5711  df-iota 6525  df-fv 6581

This theorem is referenced by:  fvssunirn  6953  elrnustOLD  24254  ustbas  24257  utopval  24262  tusval  24295  ucnval  24307  iscfilu  24318  metuval  24583  metidval  33836  pstmval  33841  measbasedom  34166  sxbrsigalem0  34236