Theorem elfvex 5660

Description: If a function value is inhabited, the function value is a set. (Contributed by Jim Kingdon, 30-Jan-2026.)

Assertion

Ref	Expression
elfvex	⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ∈ V)

Proof of Theorem elfvex

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5325	. 2 ⊢ (𝐹‘𝐵) = (℩𝑤𝐵𝐹𝑤)
2		eliotaeu 5306	. . . 4 ⊢ (𝐴 ∈ (℩𝑤𝐵𝐹𝑤) → ∃!𝑤 𝐵𝐹𝑤)
3	2, 1	eleq2s 2324	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃!𝑤 𝐵𝐹𝑤)
4		euiotaex 5294	. . 3 ⊢ (∃!𝑤 𝐵𝐹𝑤 → (℩𝑤𝐵𝐹𝑤) ∈ V)
5	3, 4	syl 14	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (℩𝑤𝐵𝐹𝑤) ∈ V)
6	1, 5	eqeltrid 2316	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ∈ V)

Syntax hints:   → wi 4  ∃!weu 2077   ∈ wcel 2200  Vcvv 2799   class class class wbr 4082  ℩cio 5275  ‘cfv 5317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3888  df-iota 5277  df-fv 5325

This theorem is referenced by:  fvmbr  5661  wlkvtxiedgg  16042