Theorem riota1 6995

Description: Property of restricted iota. Compare iota1 6203. (Contributed by Mario Carneiro, 15-Oct-2016.)

Assertion

Ref	Expression
riota1	⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riota1

Step	Hyp	Ref	Expression
1		df-reu 3112	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		iota1 6203	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
3	1, 2	sylbi 218	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
4		df-riota 6977	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	eqeq1i 2800	. 2 ⊢ ((℩𝑥 ∈ 𝐴 𝜑) = 𝑥 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥)
6	3, 5	syl6bbr 290	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∃!weu 2611  ∃!wreu 3107  ℩cio 6187  ℩crio 6976

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-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-rex 3111  df-reu 3112  df-v 3439  df-sbc 3707  df-un 3864  df-sn 4473  df-pr 4475  df-uni 4746  df-iota 6189  df-riota 6977

This theorem is referenced by:  nosupbnd1  32824  nosupbnd2  32826  wessf1ornlem  41004  disjinfi  41013