Theorem fvmbr 5674

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 5334	. . 3 ⊢ (𝐹‘𝑋) = (℩𝑤𝑋𝐹𝑤)
2	1	eqcomi 2235	. 2 ⊢ (℩𝑤𝑋𝐹𝑤) = (𝐹‘𝑋)
3		elfvfvex 5673	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝑋) → (𝐹‘𝑋) ∈ V)
4		eliotaeu 5315	. . . 4 ⊢ (𝐴 ∈ (℩𝑤𝑋𝐹𝑤) → ∃!𝑤 𝑋𝐹𝑤)
5	4, 1	eleq2s 2326	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝑋) → ∃!𝑤 𝑋𝐹𝑤)
6		breq2 4092	. . . 4 ⊢ (𝑤 = (𝐹‘𝑋) → (𝑋𝐹𝑤 ↔ 𝑋𝐹(𝐹‘𝑋)))
7	6	iota2 5316	. . 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 1397  ∃!weu 2079   ∈ wcel 2202  Vcvv 2802   class class class wbr 4088  ℩cio 5284  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334

This theorem is referenced by:  wlkvtxiedg  16195  wlkvtxiedgg  16196