Theorem riotass 7400

Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
riotass	⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotass

Step	Hyp	Ref	Expression
1		reuss 4279	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑)
2		riotasbc 7387	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
3	1, 2	syl 18	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
4		simp1 1153	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → 𝐴 ⊆ 𝐵)
5		riotacl 7386	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
6	1, 5	syl 18	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
7	4, 6	sseldd 3937	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵)
8		simp3 1155	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐵 𝜑)
9		nfriota1 7376	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
10	9	nfsbc1 3762	. . . . 5 ⊢ Ⅎ𝑥[(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑
11		sbceq1a 3754	. . . . 5 ⊢ (𝑥 = (℩𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑))
12	9, 10, 11	riota2f 7393	. . . 4 ⊢ (((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑)))
13	7, 8, 12	syl2anc 595	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑)))
14	3, 13	mpbid 235	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑))
15	14	eqcomd 2768	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  ∃!wreu 3366  [wsbc 3743   ⊆ wss 3904  ℩crio 7368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-un 3909  df-ss 3921  df-sn 4589  df-pr 4591  df-uni 4872  df-iota 6492  df-riota 7369

This theorem is used by:  moriotass  7401  resubeqsub  43219