Theorem mclsind 36421

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 36414	. 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 4137	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (𝐸 ∩ 𝑄))
14		mclsax.v	. . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇)
15	14, 2, 7	mvhf 36409	. . . . . . . . . 10 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
16	4, 15	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
17	16	ffnd 6710	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn 𝑉)
18	16	ffvelcdmda 7084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝐸)
19		mclsind.5	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝑄)
20	18, 19	elind 4097	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄))
21	20	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄))
22		ffnfv 7119	. . . . . . . 8 ⊢ (𝐻:𝑉⟶(𝐸 ∩ 𝑄) ↔ (𝐻 Fn 𝑉 ∧ ∀𝑣 ∈ 𝑉 (𝐻‘𝑣) ∈ (𝐸 ∩ 𝑄)))
23	17, 21, 22	sylanbrc 595	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑉⟶(𝐸 ∩ 𝑄))
24	23	frnd 6718	. . . . . 6 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐸 ∩ 𝑄))
25	13, 24	unssd 4190	. . . . 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 4134	. . . . . . . . . . . 12 ⊢ (𝐸 ∩ 𝑄) ⊆ 𝑄
28	26, 27	sstrdi 3920	. . . . . . . . . . 11 ⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)
29	4	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑇 ∈ mFS)
30		eqid 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
31	14, 30, 9, 2	msubff 36381	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ mFS → 𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))
32		frn 6717	. . . . . . . . . . . . . . . . . . . 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 3909	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ (𝐸 ↑m 𝐸))
36		elmapi 8872	. . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
39	8, 38	maxsta 36405	. . . . . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
42	41, 38	mstapst 36398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
43	40, 42	sstrdi 3920	. . . . . . . . . . . . . . . . . . 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 3909	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
46	1, 2, 41	elmpst 36387	. . . . . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . . 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 4097	. . . . . . . . . . . . . 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 423	. . . . . . . . . . 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 416	. . . . . . . . 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 3154	. . . . . . . 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 418	. . . . . . 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 1959	. . . . . 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 1960	. . . . 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 6899	. . . . . . 7 ⊢ 𝐸 ∈ V
63	62	inex1 5276	. . . . . 6 ⊢ (𝐸 ∩ 𝑄) ∈ V
64		sseq2 3934	. . . . . . 7 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ (𝐸 ∩ 𝑄)))
65		sseq2 3934	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝐸 ∩ 𝑄) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸 ∩ 𝑄)))
66	65	anbi1d 643	. . . . . . . . . . . 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 2849	. . . . . . . . . . . 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 3185	. . . . . . . . . 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 1952	. . . . . . . 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 1955	. . . . . . 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 644	. . . . . 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 3632	. . . . 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 595	. . . 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 4922	. . . 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 3920	. 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 3942	1 ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  ⟨cotp 4857  ∩ cint 4906   class class class wbr 5102   × cxp 5648  ^◡ccnv 5649  ran crn 5651   “ cima 5653   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8850   ↑_pm cpm 8851  Fincfn 8976  mVRcmvar 36312  mAxcmax 36316  mRExcmrex 36317  mExcmex 36318  mDVcmdv 36319  mVarscmvrs 36320  mSubstcmsub 36322  mVHcmvh 36323  mPreStcmpst 36324  mStatcmsta 36326  mFScmfs 36327  mClscmcls 36328

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-map 8852  df-pm 8853  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-n0 12610  df-z 12697  df-uz 12969  df-fz 13643  df-fzo 13792  df-seq 14148  df-hash 14478  df-word 14662  df-concat 14719  df-s1 14746  df-struct 17330  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-0g 17617  df-gsum 17618  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-submnd 18984  df-frmd 19050  df-mrex 36337  df-mex 36338  df-mrsub 36341  df-msub 36342  df-mvh 36343  df-mpst 36344  df-msr 36345  df-msta 36346  df-mfs 36347  df-mcls 36348

This theorem is used by:  mclspps  36435