Theorem iotain 44449

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 2569	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		vex 3440	. . . . 5 ⊢ 𝑦 ∈ V
3	2	intsn 4934	. . . 4 ⊢ ∩ {𝑦} = 𝑦
4		abbi 2796	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
5		df-sn 4577	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
6	4, 5	eqtr4di 2784	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
7	6	inteqd 4902	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦})
8		iotaval 6455	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
9	3, 7, 8	3eqtr4a 2792	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
10	9	exlimiv 1931	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
11	1, 10	sylbi 217	1 ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541  ∃wex 1780  ∃!weu 2563  {cab 2709  {csn 4576  ∩ cint 4897  ℩cio 6435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-v 3438  df-un 3907  df-in 3909  df-ss 3919  df-sn 4577  df-pr 4579  df-uni 4860  df-int 4898  df-iota 6437

This theorem is referenced by: (None)