Theorem mclsind 35752

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 35745	. 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 4181	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (𝐸 ∩ 𝑄))
14		mclsax.v	. . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇)
15	14, 2, 7	mvhf 35740	. . . . . . . . . 10 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
16	4, 15	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
17	16	ffnd 6669	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn 𝑉)
18	16	ffvelcdmda 7036	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝐸)
19		mclsind.5	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝑄)
20	18, 19	elind 4140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄))
21	20	ralrimiva 3129	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄))
22		ffnfv 7071	. . . . . . . 8 ⊢ (𝐻:𝑉⟶(𝐸 ∩ 𝑄) ↔ (𝐻 Fn 𝑉 ∧ ∀𝑣 ∈ 𝑉 (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄)))
23	17, 21, 22	sylanbrc 584	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑉⟶(𝐸 ∩ 𝑄))
24	23	frnd 6676	. . . . . 6 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐸 ∩ 𝑄))
25	13, 24	unssd 4132	. . . . 5 ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐸 ∩ 𝑄))
26		id 22	. . . . . . . . . . . 12 ⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄))
27		inss2 4178	. . . . . . . . . . . 12 ⊢ (𝐸 ∩ 𝑄) ⊆ 𝑄
28	26, 27	sstrdi 3934	. . . . . . . . . . 11 ⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)
29	4	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑇 ∈ mFS)
30		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
31	14, 30, 9, 2	msubff 35712	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ mFS → 𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))
32		frn 6675	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸 ↑m 𝐸) → ran 𝐿 ⊆ (𝐸 ↑m 𝐸))
33	29, 31, 32	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ran 𝐿 ⊆ (𝐸 ↑m 𝐸))
34		simpr2 1197	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ ran 𝐿)
35	33, 34	sseldd 3922	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ (𝐸 ↑m 𝐸))
36		elmapi 8796	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ (𝐸 ↑m 𝐸) → 𝑠:𝐸⟶𝐸)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠:𝐸⟶𝐸)
38		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
39	8, 38	maxsta 35736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
40	29, 39	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mStat‘𝑇))
41		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
42	41, 38	mstapst 35729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
43	40, 42	sstrdi 3934	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mPreSt‘𝑇))
44		simpr1 1196	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴)
45	43, 44	sseldd 3922	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
46	1, 2, 41	elmpst 35718	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
47	46	simp3bi 1148	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) → 𝑝 ∈ 𝐸)
48	45, 47	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑝 ∈ 𝐸)
49	37, 48	ffvelcdmd 7037	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → (𝑠‘𝑝) ∈ 𝐸)
50	49	3adant3 1133	. . . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . . 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 1120	. . . . . . . . . . . . 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 1355	. . . . . . . . . . . 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 417	. . . . . . . . . . 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 34	. . . . . . . . . 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 410	. . . . . . . . 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 3129	. . . . . . . 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 412	. . . . . . 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 1929	. . . . . 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 1930	. . . . 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 6854	. . . . . . 7 ⊢ 𝐸 ∈ V
63	62	inex1 5258	. . . . . 6 ⊢ (𝐸 ∩ 𝑄) ∈ V
64		sseq2 3948	. . . . . . 7 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ (𝐸 ∩ 𝑄)))
65		sseq2 3948	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄)))
66	65	anbi1d 632	. . . . . . . . . . . 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 2825	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝑠‘𝑝) ∈ 𝑐 ↔ (𝑠‘𝑝) ∈ (𝐸 ∩ 𝑄)))
68	66, 67	imbi12d 344	. . . . . . . . . . 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 3160	. . . . . . . . . 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 340	. . . . . . . . 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 1922	. . . . . . . 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 1925	. . . . . . 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 633	. . . . . 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 3622	. . . . 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 584	. . . 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 4905	. . . 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 17	. . 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 3934	. 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 3956	1 ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ⟨cotp 4575  ∩ cint 4889   class class class wbr 5085   × cxp 5629  ^◡ccnv 5630  ran crn 5632   “ cima 5634   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773   ↑_pm cpm 8774  Fincfn 8893  mVRcmvar 35643  mAxcmax 35647  mRExcmrex 35648  mExcmex 35649  mDVcmdv 35650  mVarscmvrs 35651  mSubstcmsub 35653  mVHcmvh 35654  mPreStcmpst 35655  mStatcmsta 35657  mFScmfs 35658  mClscmcls 35659

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-word 14476  df-concat 14533  df-s1 14559  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-gsum 17405  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-frmd 18817  df-mrex 35668  df-mex 35669  df-mrsub 35672  df-msub 35673  df-mvh 35674  df-mpst 35675  df-msr 35676  df-msta 35677  df-mfs 35678  df-mcls 35679

This theorem is referenced by:  mclspps  35766