Theorem riotass2 7399

Description: Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)

Assertion

Ref	Expression
riotass2	⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem riotass2

Step	Hyp	Ref	Expression
1		reuss2 4278	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑)
2		simplr 780	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
3		riotasbc 7387	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
4		riotacl 7386	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
5		rspsbc 3831	. . . . . . 7 ⊢ ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥](𝜑 → 𝜓)))
6		sbcimg 3791	. . . . . . 7 ⊢ ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥](𝜑 → 𝜓) ↔ ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓)))
7	5, 6	sylibd 242	. . . . . 6 ⊢ ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓)))
8	4, 7	syl 18	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓)))
9	3, 8	mpid 45	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓))
10	1, 2, 9	sylc 66	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓)
11	1, 4	syl 18	. . . . 5 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
12		ssel 3930	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵))
13	12	ad2antrr 738	. . . . 5 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵))
14	11, 13	mpd 16	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵)
15		simprr 784	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ∃!𝑥 ∈ 𝐵 𝜓)
16		nfriota1 7376	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
17	16	nfsbc1 3762	. . . . 5 ⊢ Ⅎ𝑥[(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓
18		sbceq1a 3754	. . . . 5 ⊢ (𝑥 = (℩𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓))
19	16, 17, 18	riota2f 7393	. . . 4 ⊢ (((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓 ↔ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥 ∈ 𝐴 𝜑)))
20	14, 15, 19	syl2anc 595	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓 ↔ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥 ∈ 𝐴 𝜑)))
21	10, 20	mpbid 235	. 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥 ∈ 𝐴 𝜑))
22	21	eqcomd 2768	1 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃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:  fisupcl  9428  quotlem  26472  adjbdln  32446  rexdiv  33256  cdlemefrs32fva  41202  addinvcom  43221