Theorem mclsind 36040

Description: Induction theorem for closure: any other set 𝑄 closed under the axioms and the hypotheses contains all the elements of the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mclsval.d	⊢ 𝐷 = (mDV‘𝑇)
mclsval.e	⊢ 𝐸 = (mEx‘𝑇)
mclsval.c	⊢ 𝐶 = (mCls‘𝑇)
mclsval.1	⊢ (𝜑 → 𝑇 ∈ mFS)
mclsval.2	⊢ (𝜑 → 𝐾 ⊆ 𝐷)
mclsval.3	⊢ (𝜑 → 𝐵 ⊆ 𝐸)
mclsax.a	⊢ 𝐴 = (mAx‘𝑇)
mclsax.l	⊢ 𝐿 = (mSubst‘𝑇)
mclsax.v	⊢ 𝑉 = (mVR‘𝑇)
mclsax.h	⊢ 𝐻 = (mVH‘𝑇)
mclsax.w	⊢ 𝑊 = (mVars‘𝑇)
mclsind.4	⊢ (𝜑 → 𝐵 ⊆ 𝑄)
mclsind.5	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝑄)
mclsind.6	⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑄)

Assertion

Ref	Expression
mclsind	⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)

Distinct variable groups:   𝑚,𝑜,𝑝,𝑠,𝑣,𝐸   𝑥,𝑚,𝐻,𝑜,𝑝,𝑠,𝑣   𝑦,𝑚,𝐵,𝑜,𝑝,𝑠,𝑣,𝑥   𝐶,𝑚,𝑜,𝑝,𝑠,𝑣,𝑥   𝑚,𝐿,𝑜,𝑝,𝑠,𝑥,𝑦   𝐴,𝑚,𝑜,𝑝,𝑠   𝑇,𝑚,𝑜,𝑝,𝑠,𝑥,𝑦   𝜑,𝑚,𝑜,𝑝,𝑠,𝑣,𝑥,𝑦   𝑄,𝑚,𝑜,𝑝,𝑠,𝑣   𝑣,𝑉,𝑥   𝑚,𝑊,𝑜,𝑝,𝑠,𝑥   𝑚,𝐾,𝑜,𝑝,𝑠,𝑣,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣)   𝐶(𝑦)   𝐷(𝑥,𝑦,𝑣,𝑚,𝑜,𝑠,𝑝)   𝑄(𝑥,𝑦)   𝑇(𝑣)   𝐸(𝑥,𝑦)   𝐻(𝑦)   𝐿(𝑣)   𝑉(𝑦,𝑚,𝑜,𝑠,𝑝)   𝑊(𝑦,𝑣)

Proof of Theorem mclsind

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mclsval.d	. . 3 ⊢ 𝐷 = (mDV‘𝑇)
2		mclsval.e	. . 3 ⊢ 𝐸 = (mEx‘𝑇)
3		mclsval.c	. . 3 ⊢ 𝐶 = (mCls‘𝑇)
4		mclsval.1	. . 3 ⊢ (𝜑 → 𝑇 ∈ mFS)
5		mclsval.2	. . 3 ⊢ (𝜑 → 𝐾 ⊆ 𝐷)
6		mclsval.3	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
7		mclsax.h	. . 3 ⊢ 𝐻 = (mVH‘𝑇)
8		mclsax.a	. . 3 ⊢ 𝐴 = (mAx‘𝑇)
9		mclsax.l	. . 3 ⊢ 𝐿 = (mSubst‘𝑇)
10		mclsax.w	. . 3 ⊢ 𝑊 = (mVars‘𝑇)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	mclsval 36033	. 2 ⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
12		mclsind.4	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝑄)
13	6, 12	ssind 4194	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (𝐸 ∩ 𝑄))
14		mclsax.v	. . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇)
15	14, 2, 7	mvhf 36028	. . . . . . . . . 10 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
16	4, 15	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
17	16	ffnd 6708	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn 𝑉)
18	16	ffvelcdmda 7081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝐸)
19		mclsind.5	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝑄)
20	18, 19	elind 4154	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄))
21	20	ralrimiva 3157	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄))
22		ffnfv 7116	. . . . . . . 8 ⊢ (𝐻:𝑉⟶(𝐸 ∩ 𝑄) ↔ (𝐻 Fn 𝑉 ∧ ∀𝑣 ∈ 𝑉 (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄)))
23	17, 21, 22	sylanbrc 594	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑉⟶(𝐸 ∩ 𝑄))
24	23	frnd 6716	. . . . . 6 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐸 ∩ 𝑄))
25	13, 24	unssd 4146	. . . . 5 ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐸 ∩ 𝑄))
26		id 23	. . . . . . . . . . . 12 ⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄))
27		inss2 4191	. . . . . . . . . . . 12 ⊢ (𝐸 ∩ 𝑄) ⊆ 𝑄
28	26, 27	sstrdi 3950	. . . . . . . . . . 11 ⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)
29	4	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑇 ∈ mFS)
30		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
31	14, 30, 9, 2	msubff 36000	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ mFS → 𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))
32		frn 6715	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸 ↑m 𝐸) → ran 𝐿 ⊆ (𝐸 ↑m 𝐸))
33	29, 31, 32	3syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ran 𝐿 ⊆ (𝐸 ↑m 𝐸))
34		simpr2 1214	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ ran 𝐿)
35	33, 34	sseldd 3939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ (𝐸 ↑m 𝐸))
36		elmapi 8847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ (𝐸 ↑m 𝐸) → 𝑠:𝐸⟶𝐸)
37	35, 36	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠:𝐸⟶𝐸)
38		eqid 2763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
39	8, 38	maxsta 36024	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
40	29, 39	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mStat‘𝑇))
41		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
42	41, 38	mstapst 36017	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
43	40, 42	sstrdi 3950	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mPreSt‘𝑇))
44		simpr1 1213	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴)
45	43, 44	sseldd 3939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
46	1, 2, 41	elmpst 36006	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
47	46	simp3bi 1165	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) → 𝑝 ∈ 𝐸)
48	45, 47	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑝 ∈ 𝐸)
49	37, 48	ffvelcdmd 7082	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → (𝑠‘𝑝) ∈ 𝐸)
50	49	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)
51		mclsind.6	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑄)
52	50, 51	elind 4154	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))
53	52	3exp 1137	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))
54	53	3expd 1372	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → (𝑠 ∈ ran 𝐿 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄 → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))))
55	54	imp31 422	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) ∧ 𝑠 ∈ ran 𝐿) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄 → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))
56	28, 55	syl5 35	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) ∧ 𝑠 ∈ ran 𝐿) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))
57	56	impd 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) ∧ 𝑠 ∈ ran 𝐿) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))
58	57	ralrimiva 3157	. . . . . . . 8 ⊢ ((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))
59	58	ex 417	. . . . . . 7 ⊢ (𝜑 → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))
60	59	alrimiv 1957	. . . . . 6 ⊢ (𝜑 → ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))
61	60	alrimivv 1958	. . . . 5 ⊢ (𝜑 → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))
62	2	fvexi 6897	. . . . . . 7 ⊢ 𝐸 ∈ V
63	62	inex1 5287	. . . . . 6 ⊢ (𝐸 ∩ 𝑄) ∈ V
64		sseq2 3964	. . . . . . 7 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ (𝐸 ∩ 𝑄)))
65		sseq2 3964	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄)))
66	65	anbi1d 642	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) ↔ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾))))
67		eleq2 2852	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝑠‘𝑝) ∈ 𝑐 ↔ (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))
68	66, 67	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))
69	68	ralbidv 3188	. . . . . . . . . 10 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))
70	69	imbi2d 343	. . . . . . . . 9 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))))
71	70	albidv 1950	. . . . . . . 8 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))))
72	71	2albidv 1953	. . . . . . 7 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))))
73	64, 72	anbi12d 643	. . . . . 6 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) ↔ ((𝐵 ∪ ran 𝐻) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄))))))
74	63, 73	elab 3639	. . . . 5 ⊢ ((𝐸 ∩ 𝑄) ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ↔ ((𝐵 ∪ ran 𝐻) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))))
75	25, 61, 74	sylanbrc 594	. . . 4 ⊢ (𝜑 → (𝐸 ∩ 𝑄) ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
76		intss1 4929	. . . 4 ⊢ ((𝐸 ∩ 𝑄) ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ (𝐸 ∩ 𝑄))
77	75, 76	syl 18	. . 3 ⊢ (𝜑 → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ (𝐸 ∩ 𝑄))
78	77, 27	sstrdi 3950	. 2 ⊢ (𝜑 → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝑄)
79	11, 78	eqsstrd 3972	1 ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ⟨cotp 4598  ∩ cint 4913   class class class wbr 5110   × cxp 5661  ^◡ccnv 5662  ran crn 5664   “ cima 5666   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ↑_m cmap 8825   ↑_pm cpm 8826  Fincfn 8944  mVRcmvar 35931  mAxcmax 35935  mRExcmrex 35936  mExcmex 35937  mDVcmdv 35938  mVarscmvrs 35939  mSubstcmsub 35941  mVHcmvh 35942  mPreStcmpst 35943  mStatcmsta 35945  mFScmfs 35946  mClscmcls 35947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-word 14553  df-concat 14610  df-s1 14636  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-0g 17495  df-gsum 17496  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-submnd 18843  df-frmd 18909  df-mrex 35956  df-mex 35957  df-mrsub 35960  df-msub 35961  df-mvh 35962  df-mpst 35963  df-msr 35964  df-msta 35965  df-mfs 35966  df-mcls 35967

This theorem is referenced by:  mclspps  36054