Theorem elfvunirn 6924

Description: A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6929). (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 4330	. . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ≠ ∅)
2		fvn0fvelrn 6923	. . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹‘𝐴) ∈ ran 𝐹)
3		elssuni 4935	. . . 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 3971	. 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 2098   ≠ wne 2930   ⊆ wss 3939  ∅c0 4318  ∪ cuni 4903  ran crn 5673  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-cnv 5680  df-dm 5682  df-rn 5683  df-iota 6495  df-fv 6551

This theorem is referenced by:  fvssunirn  6925  elrnustOLD  24147  ustbas  24150  utopval  24155  tusval  24188  ucnval  24200  iscfilu  24211  metuval  24476  metidval  33548  pstmval  33553  measbasedom  33878  sxbrsigalem0  33948