Theorem riotass2 7146

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 4285	. . . 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 7134	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)
4		riotacl 7133	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
5		rspsbc 3864	. . . . . . 7 ⊢ ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥](𝜑 → 𝜓)))
6		sbcimg 3822	. . . . . . 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 241	. . . . . 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 3963	. . . . . 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 7123	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
17	16	nfsbc1 3793	. . . . 5 ⊢ Ⅎ𝑥[(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓
18		sbceq1a 3785	. . . . 5 ⊢ (𝑥 = (℩𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓))
19	16, 17, 18	riota2f 7140	. . . 4 ⊢ (((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜓 ↔ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥 ∈ 𝐴 𝜑)))
20	14, 15, 19	syl2anc 586	. . 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 234	. 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥 ∈ 𝐴 𝜑))
22	21	eqcomd 2829	1 ⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  ∃!wreu 3142  [wsbc 3774   ⊆ wss 3938  ℩crio 7115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-un 3943  df-in 3945  df-ss 3954  df-sn 4570  df-pr 4572  df-uni 4841  df-iota 6316  df-riota 7116

This theorem is referenced by:  fisupcl  8935  quotlem  24891  adjbdln  29862  rexdiv  30604  cdlemefrs32fva  37538