Theorem elfvfvex 5682

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 5341	. 2 ⊢ (𝐹‘𝐵) = (℩𝑤𝐵𝐹𝑤)
2		eliotaeu 5322	. . . 4 ⊢ (𝐴 ∈ (℩𝑤𝐵𝐹𝑤) → ∃!𝑤 𝐵𝐹𝑤)
3	2, 1	eleq2s 2326	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃!𝑤 𝐵𝐹𝑤)
4		euiotaex 5310	. . 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 2803   class class class wbr 4093  ℩cio 5291  ‘cfv 5333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-sn 3679  df-pr 3680  df-uni 3899  df-iota 5293  df-fv 5341

This theorem is referenced by:  fvmbr  5683  wlkvtxiedg  16269  wlkvtxiedgg  16270