Theorem fvmbr 5705

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 5360	. . 3 ⊢ (𝐹‘𝑋) = (℩𝑤𝑋𝐹𝑤)
2	1	eqcomi 2236	. 2 ⊢ (℩𝑤𝑋𝐹𝑤) = (𝐹‘𝑋)
3		elfvfvex 5704	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝑋) → (𝐹‘𝑋) ∈ V)
4		eliotaeu 5341	. . . 4 ⊢ (𝐴 ∈ (℩𝑤𝑋𝐹𝑤) → ∃!𝑤 𝑋𝐹𝑤)
5	4, 1	eleq2s 2327	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝑋) → ∃!𝑤 𝑋𝐹𝑤)
6		breq2 4113	. . . 4 ⊢ (𝑤 = (𝐹‘𝑋) → (𝑋𝐹𝑤 ↔ 𝑋𝐹(𝐹‘𝑋)))
7	6	iota2 5342	. . 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 1398  ∃!weu 2080   ∈ wcel 2203  Vcvv 2813   class class class wbr 4109  ℩cio 5310  ‘cfv 5352

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360

This theorem is referenced by:  wlkvtxiedg  16340  wlkvtxiedgg  16341