Theorem elfvfvex 5673

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

Assertion

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

Proof of Theorem elfvfvex

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5334	. 2 ⊢ (𝐹‘𝐵) = (℩𝑤𝐵𝐹𝑤)
2		eliotaeu 5315	. . . 4 ⊢ (𝐴 ∈ (℩𝑤𝐵𝐹𝑤) → ∃!𝑤 𝐵𝐹𝑤)
3	2, 1	eleq2s 2326	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃!𝑤 𝐵𝐹𝑤)
4		euiotaex 5303	. . 3 ⊢ (∃!𝑤 𝐵𝐹𝑤 → (℩𝑤𝐵𝐹𝑤) ∈ V)
5	3, 4	syl 14	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (℩𝑤𝐵𝐹𝑤) ∈ V)
6	1, 5	eqeltrid 2318	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ∈ V)

Syntax hints:   → wi 4  ∃!weu 2079   ∈ wcel 2202  Vcvv 2802   class class class wbr 4088  ℩cio 5284  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-sn 3675  df-pr 3676  df-uni 3894  df-iota 5286  df-fv 5334

This theorem is referenced by:  fvmbr  5674  wlkvtxiedg  16195  wlkvtxiedgg  16196