Theorem fvmbr 5725

Description: If a function value is inhabited, the argument is related to the function value. (Contributed by Jim Kingdon, 31-Jan-2026.)

Assertion

Ref	Expression
fvmbr	⊢ (𝐴 ∈ (𝐹‘𝑋) → 𝑋𝐹(𝐹‘𝑋))

Proof of Theorem fvmbr

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5380	. . 3 ⊢ (𝐹‘𝑋) = (℩𝑤𝑋𝐹𝑤)
2	1	eqcomi 2242	. 2 ⊢ (℩𝑤𝑋𝐹𝑤) = (𝐹‘𝑋)
3		elfvfvex 5724	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝑋) → (𝐹‘𝑋) ∈ V)
4		eliotaeu 5361	. . . 4 ⊢ (𝐴 ∈ (℩𝑤𝑋𝐹𝑤) → ∃!𝑤 𝑋𝐹𝑤)
5	4, 1	eleq2s 2333	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝑋) → ∃!𝑤 𝑋𝐹𝑤)
6		breq2 4129	. . . 4 ⊢ (𝑤 = (𝐹‘𝑋) → (𝑋𝐹𝑤 ↔ 𝑋𝐹(𝐹‘𝑋)))
7	6	iota2 5362	. . 3 ⊢ (((𝐹‘𝑋) ∈ V ∧ ∃!𝑤 𝑋𝐹𝑤) → (𝑋𝐹(𝐹‘𝑋) ↔ (℩𝑤𝑋𝐹𝑤) = (𝐹‘𝑋)))
8	3, 5, 7	syl2anc 415	. 2 ⊢ (𝐴 ∈ (𝐹‘𝑋) → (𝑋𝐹(𝐹‘𝑋) ↔ (℩𝑤𝑋𝐹𝑤) = (𝐹‘𝑋)))
9	2, 8	mpbiri 168	1 ⊢ (𝐴 ∈ (𝐹‘𝑋) → 𝑋𝐹(𝐹‘𝑋))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  ∃!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-3an 1011  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-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380

This theorem is referenced by:  wlkvtxiedg  16500  wlkvtxiedgg  16501