Theorem iotain 40287

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 2617	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		vex 3440	. . . . 5 ⊢ 𝑦 ∈ V
3	2	intsn 4818	. . . 4 ⊢ ∩ {𝑦} = 𝑦
4		abbi1 2859	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
5		df-sn 4473	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
6	4, 5	syl6eqr 2849	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
7	6	inteqd 4787	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦})
8		iotaval 6200	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
9	3, 7, 8	3eqtr4a 2857	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
10	9	exlimiv 1908	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
11	1, 10	sylbi 218	1 ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1520   = wceq 1522  ∃wex 1761  ∃!weu 2611  {cab 2775  {csn 4472  ∩ cint 4782  ℩cio 6187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-un 3864  df-in 3866  df-sn 4473  df-pr 4475  df-uni 4746  df-int 4783  df-iota 6189

This theorem is referenced by: (None)