Theorem iotain 42035

Description: Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 15-Jul-2011.)

Assertion

Ref	Expression
iotain	⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))

Proof of Theorem iotain

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eu6 2574	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
3	2	intsn 4917	. . . 4 ⊢ ∩ {𝑦} = 𝑦
4		abbi1 2806	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
5		df-sn 4562	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
6	4, 5	eqtr4di 2796	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
7	6	inteqd 4884	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦})
8		iotaval 6407	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
9	3, 7, 8	3eqtr4a 2804	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
10	9	exlimiv 1933	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
11	1, 10	sylbi 216	1 ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  ∃wex 1782  ∃!weu 2568  {cab 2715  {csn 4561  ∩ cint 4879  ℩cio 6389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564  df-uni 4840  df-int 4880  df-iota 6391

This theorem is referenced by: (None)