Theorem elfvunirn 6939

Description: A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6944). (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 4347	. . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ≠ ∅)
2		fvn0fvelrn 6938	. . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹‘𝐴) ∈ ran 𝐹)
3		elssuni 4942	. . . 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 3994	. 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 2106   ≠ wne 2938   ⊆ wss 3963  ∅c0 4339  ∪ cuni 4912  ran crn 5690  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571

This theorem is referenced by:  fvssunirn  6940  elrnustOLD  24249  ustbas  24252  utopval  24257  tusval  24290  ucnval  24302  iscfilu  24313  metuval  24578  metidval  33851  pstmval  33856  measbasedom  34183  sxbrsigalem0  34253