Theorem elfvfvex 5724

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 5380	. 2 ⊢ (𝐹‘𝐵) = (℩𝑤𝐵𝐹𝑤)
2		eliotaeu 5361	. . . 4 ⊢ (𝐴 ∈ (℩𝑤𝐵𝐹𝑤) → ∃!𝑤 𝐵𝐹𝑤)
3	2, 1	eleq2s 2333	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃!𝑤 𝐵𝐹𝑤)
4		euiotaex 5349	. . 3 ⊢ (∃!𝑤 𝐵𝐹𝑤 → (℩𝑤𝐵𝐹𝑤) ∈ V)
5	3, 4	syl 14	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (℩𝑤𝐵𝐹𝑤) ∈ V)
6	1, 5	eqeltrid 2325	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → (𝐹‘𝐵) ∈ V)

Syntax hints:   → wi 4  ∃!weu 2086   ∈ wcel 2209  Vcvv 2821   class class class wbr 4125  ℩cio 5330  ‘cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3711  df-pr 3712  df-uni 3931  df-iota 5332  df-fv 5380

This theorem is referenced by:  fvmbr  5725  wlkvtxiedg  16500  wlkvtxiedgg  16501