Theorem riotass 6538

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 3867	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑)
2		riotasbc 6526	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
4		simp1 1054	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → 𝐴 ⊆ 𝐵)
5		riotacl 6525	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
6	1, 5	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
7	4, 6	sseldd 3569	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵)
8		simp3 1056	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐵 𝜑)
9		nfriota1 6518	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
10	9	nfsbc1 3421	. . . . 5 ⊢ Ⅎ𝑥[(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑
11		sbceq1a 3413	. . . . 5 ⊢ (𝑥 = (℩𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑))
12	9, 10, 11	riota2f 6532	. . . 4 ⊢ (((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑)))
13	7, 8, 12	syl2anc 691	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑)))
14	3, 13	mpbid 221	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑))
15	14	eqcomd 2616	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ∃!wreu 2898  [wsbc 3402   ⊆ wss 3540  ℩crio 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-riota 6511

This theorem is referenced by:  moriotass  6539