Theorem reiotacl2 4364

Description: Membership law for descriptions. (Contributed by SF, 21-Aug-2011.)

Assertion

Ref	Expression
reiotacl2	⊢ (∃!x ∈ A φ → (℩x(x ∈ A ∧ φ)) ∈ {x ∈ A ∣ φ})

Proof of Theorem reiotacl2

Step	Hyp	Ref	Expression
1		df-reu 2622	. . 3 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
2		iotacl 4363	. . 3 ⊢ (∃!x(x ∈ A ∧ φ) → (℩x(x ∈ A ∧ φ)) ∈ {x ∣ (x ∈ A ∧ φ)})
3	1, 2	sylbi 187	. 2 ⊢ (∃!x ∈ A φ → (℩x(x ∈ A ∧ φ)) ∈ {x ∣ (x ∈ A ∧ φ)})
4		df-rab 2624	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
5	3, 4	syl6eleqr 2444	1 ⊢ (∃!x ∈ A φ → (℩x(x ∈ A ∧ φ)) ∈ {x ∈ A ∣ φ})

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∃!weu 2204  {cab 2339  ∃!wreu 2617  {crab 2619  ℩cio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-reu 2622  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by:  reiotacl  4365