Theorem fvmbr 5661

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 5325	. . 3 ⊢ (𝐹‘𝑋) = (℩𝑤𝑋𝐹𝑤)
2	1	eqcomi 2233	. 2 ⊢ (℩𝑤𝑋𝐹𝑤) = (𝐹‘𝑋)
3		elfvex 5660	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝑋) → (𝐹‘𝑋) ∈ V)
4		eliotaeu 5306	. . . 4 ⊢ (𝐴 ∈ (℩𝑤𝑋𝐹𝑤) → ∃!𝑤 𝑋𝐹𝑤)
5	4, 1	eleq2s 2324	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝑋) → ∃!𝑤 𝑋𝐹𝑤)
6		breq2 4086	. . . 4 ⊢ (𝑤 = (𝐹‘𝑋) → (𝑋𝐹𝑤 ↔ 𝑋𝐹(𝐹‘𝑋)))
7	6	iota2 5307	. . 3 ⊢ (((𝐹‘𝑋) ∈ V ∧ ∃!𝑤 𝑋𝐹𝑤) → (𝑋𝐹(𝐹‘𝑋) ↔ (℩𝑤𝑋𝐹𝑤) = (𝐹‘𝑋)))
8	3, 5, 7	syl2anc 411	. 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 1395  ∃!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-3an 1004  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-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325

This theorem is referenced by:  wlkvtxiedgg  16042