Theorem riotass2 7380

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 4311	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑)
2		simplr 767	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
3		riotasbc 7368	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
4		riotacl 7367	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
5		rspsbc 3869	. . . . . . 7 ⊢ ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥](𝜑 → 𝜓)))
6		sbcimg 3824	. . . . . . 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 238	. . . . . 6 ⊢ ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓)))
8	4, 7	syl 17	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓)))
9	3, 8	mpid 44	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓))
10	1, 2, 9	sylc 65	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓)
11	1, 4	syl 17	. . . . 5 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
12		ssel 3971	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵))
13	12	ad2antrr 724	. . . . 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 15	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵)
15		simprr 771	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ∃!𝑥 ∈ 𝐵 𝜓)
16		nfriota1 7356	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
17	16	nfsbc1 3792	. . . . 5 ⊢ Ⅎ𝑥[(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓
18		sbceq1a 3784	. . . . 5 ⊢ (𝑥 = (℩𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓))
19	16, 17, 18	riota2f 7374	. . . 4 ⊢ (((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓 ↔ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥 ∈ 𝐴 𝜑)))
20	14, 15, 19	syl2anc 584	. . 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 231	. 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥 ∈ 𝐴 𝜑))
22	21	eqcomd 2737	1 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  ∃!wreu 3373  [wsbc 3773   ⊆ wss 3944  ℩crio 7348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-un 3949  df-in 3951  df-ss 3961  df-sn 4623  df-pr 4625  df-uni 4902  df-iota 6484  df-riota 7349

This theorem is referenced by:  fisupcl  9446  quotlem  25742  adjbdln  31199  rexdiv  31963  cdlemefrs32fva  39076  addinvcom  41086