Theorem mclsval 33425

Description: The function mapping variables to variable expressions is one-to-one. (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)
mclsval.h	⊢ 𝐻 = (mVH‘𝑇)
mclsval.a	⊢ 𝐴 = (mAx‘𝑇)
mclsval.s	⊢ 𝑆 = (mSubst‘𝑇)
mclsval.v	⊢ 𝑉 = (mVars‘𝑇)

Assertion

Ref	Expression
mclsval	⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})

Distinct variable groups:   𝑚,𝑐,𝑜,𝑝,𝑠,𝐸   𝑥,𝑐,𝐻,𝑚,𝑜,𝑝,𝑠   𝑦,𝑐,𝐵,𝑚,𝑜,𝑝,𝑠,𝑥   𝐶,𝑚,𝑜,𝑝,𝑠,𝑥   𝐴,𝑐,𝑚,𝑜,𝑝,𝑠   𝑆,𝑐,𝑠,𝑥,𝑦   𝑇,𝑐,𝑚,𝑜,𝑝,𝑠,𝑥,𝑦   𝜑,𝑐,𝑚,𝑜,𝑝,𝑠,𝑥,𝑦   𝑉,𝑐,𝑥   𝐾,𝑐,𝑚,𝑜,𝑝,𝑠,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑦,𝑐)   𝐷(𝑥,𝑦,𝑚,𝑜,𝑠,𝑝,𝑐)   𝑆(𝑚,𝑜,𝑝)   𝐸(𝑥,𝑦)   𝐻(𝑦)   𝑉(𝑦,𝑚,𝑜,𝑠,𝑝)

Proof of Theorem mclsval

Dummy variables ℎ 𝑑 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mclsval.c	. . 3 ⊢ 𝐶 = (mCls‘𝑇)
2		mclsval.1	. . . 4 ⊢ (𝜑 → 𝑇 ∈ mFS)
3		elex 3440	. . . 4 ⊢ (𝑇 ∈ mFS → 𝑇 ∈ V)
4		fveq2 6756	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
5		mclsval.d	. . . . . . . 8 ⊢ 𝐷 = (mDV‘𝑇)
6	4, 5	eqtr4di 2797	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
7	6	pweqd 4549	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝐷)
8		fveq2 6756	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
9		mclsval.e	. . . . . . . 8 ⊢ 𝐸 = (mEx‘𝑇)
10	8, 9	eqtr4di 2797	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
11	10	pweqd 4549	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
12		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mVH‘𝑡) = (mVH‘𝑇))
13		mclsval.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (mVH‘𝑇)
14	12, 13	eqtr4di 2797	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (mVH‘𝑡) = 𝐻)
15	14	rneqd 5836	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ran (mVH‘𝑡) = ran 𝐻)
16	15	uneq2d 4093	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (ℎ ∪ ran (mVH‘𝑡)) = (ℎ ∪ ran 𝐻))
17	16	sseq1d 3948	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ↔ (ℎ ∪ ran 𝐻) ⊆ 𝑐))
18		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
19		mclsval.a	. . . . . . . . . . . . . 14 ⊢ 𝐴 = (mAx‘𝑇)
20	18, 19	eqtr4di 2797	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
21	20	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) ↔ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴))
22		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (mSubst‘𝑡) = (mSubst‘𝑇))
23		mclsval.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (mSubst‘𝑇)
24	22, 23	eqtr4di 2797	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (mSubst‘𝑡) = 𝑆)
25	24	rneqd 5836	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ran (mSubst‘𝑡) = ran 𝑆)
26	15	uneq2d 4093	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (𝑜 ∪ ran (mVH‘𝑡)) = (𝑜 ∪ ran 𝐻))
27	26	imaeq2d 5958	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → (𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) = (𝑠 “ (𝑜 ∪ ran 𝐻)))
28	27	sseq1d 3948	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → ((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
29		fveq2 6756	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
30		mclsval.v	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑉 = (mVars‘𝑇)
31	29, 30	eqtr4di 2797	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
32	14	fveq1d 6758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → ((mVH‘𝑡)‘𝑥) = (𝐻‘𝑥))
33	32	fveq2d 6760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (𝑠‘((mVH‘𝑡)‘𝑥)) = (𝑠‘(𝐻‘𝑥)))
34	31, 33	fveq12d 6763	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) = (𝑉‘(𝑠‘(𝐻‘𝑥))))
35	14	fveq1d 6758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → ((mVH‘𝑡)‘𝑦) = (𝐻‘𝑦))
36	35	fveq2d 6760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (𝑠‘((mVH‘𝑡)‘𝑦)) = (𝑠‘(𝐻‘𝑦)))
37	31, 36	fveq12d 6763	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦))) = (𝑉‘(𝑠‘(𝐻‘𝑦))))
38	34, 37	xpeq12d 5611	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) = ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))))
39	38	sseq1d 3948	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → ((((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑 ↔ ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑))
40	39	imbi2d 340	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ((𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) ↔ (𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)))
41	40	2albidv 1927	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) ↔ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)))
42	28, 41	anbi12d 630	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) ↔ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑))))
43	42	imbi1d 341	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ((((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))
44	25, 43	raleqbidv 3327	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))
45	21, 44	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐))))
46	45	albidv 1924	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐))))
47	46	2albidv 1927	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐))))
48	17, 47	anbi12d 630	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐))) ↔ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))))
49	48	abbidv 2808	. . . . . . 7 ⊢ (𝑡 = 𝑇 → {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))} = {𝑐 ∣ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))})
50	49	inteqd 4881	. . . . . 6 ⊢ (𝑡 = 𝑇 → ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))} = ∩ {𝑐 ∣ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))})
51	7, 11, 50	mpoeq123dv 7328	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}) = (𝑑 ∈ 𝒫 𝐷, ℎ ∈ 𝒫 𝐸 ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}))
52		df-mcls 33359	. . . . 5 ⊢ mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}))
53	5	fvexi 6770	. . . . . . 7 ⊢ 𝐷 ∈ V
54	53	pwex 5298	. . . . . 6 ⊢ 𝒫 𝐷 ∈ V
55	9	fvexi 6770	. . . . . . 7 ⊢ 𝐸 ∈ V
56	55	pwex 5298	. . . . . 6 ⊢ 𝒫 𝐸 ∈ V
57	54, 56	mpoex 7893	. . . . 5 ⊢ (𝑑 ∈ 𝒫 𝐷, ℎ ∈ 𝒫 𝐸 ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}) ∈ V
58	51, 52, 57	fvmpt 6857	. . . 4 ⊢ (𝑇 ∈ V → (mCls‘𝑇) = (𝑑 ∈ 𝒫 𝐷, ℎ ∈ 𝒫 𝐸 ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}))
59	2, 3, 58	3syl 18	. . 3 ⊢ (𝜑 → (mCls‘𝑇) = (𝑑 ∈ 𝒫 𝐷, ℎ ∈ 𝒫 𝐸 ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}))
60	1, 59	syl5eq 2791	. 2 ⊢ (𝜑 → 𝐶 = (𝑑 ∈ 𝒫 𝐷, ℎ ∈ 𝒫 𝐸 ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}))
61		simprr 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → ℎ = 𝐵)
62	61	uneq1d 4092	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → (ℎ ∪ ran 𝐻) = (𝐵 ∪ ran 𝐻))
63	62	sseq1d 3948	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ 𝑐))
64		simprl 767	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → 𝑑 = 𝐾)
65	64	sseq2d 3949	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → (((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑 ↔ ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾))
66	65	imbi2d 340	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → ((𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑) ↔ (𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)))
67	66	2albidv 1927	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑) ↔ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)))
68	67	anbi2d 628	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) ↔ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾))))
69	68	imbi1d 341	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))
70	69	ralbidv 3120	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → (∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))
71	70	imbi2d 340	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
72	71	albidv 1924	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
73	72	2albidv 1927	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
74	63, 73	anbi12d 630	. . . 4 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → (((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐))) ↔ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))))
75	74	abbidv 2808	. . 3 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → {𝑐 ∣ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))} = {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
76	75	inteqd 4881	. 2 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → ∩ {𝑐 ∣ ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))} = ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
77		mclsval.2	. . 3 ⊢ (𝜑 → 𝐾 ⊆ 𝐷)
78	53	elpw2 5264	. . 3 ⊢ (𝐾 ∈ 𝒫 𝐷 ↔ 𝐾 ⊆ 𝐷)
79	77, 78	sylibr 233	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝒫 𝐷)
80		mclsval.3	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
81	55	elpw2 5264	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐸 ↔ 𝐵 ⊆ 𝐸)
82	80, 81	sylibr 233	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐸)
83	5, 9, 1, 2, 77, 80, 13, 19, 23, 30	mclsssvlem 33424	. . 3 ⊢ (𝜑 → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸)
84	55	ssex 5240	. . 3 ⊢ (∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸 → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ∈ V)
85	83, 84	syl 17	. 2 ⊢ (𝜑 → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ∈ V)
86	60, 76, 79, 82, 85	ovmpod 7403	1 ⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  𝒫 cpw 4530  ⟨cotp 4566  ∩ cint 4876   class class class wbr 5070   × cxp 5578  ran crn 5581   “ cima 5583  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  mAxcmax 33327  mExcmex 33329  mDVcmdv 33330  mVarscmvrs 33331  mSubstcmsub 33333  mVHcmvh 33334  mFScmfs 33338  mClscmcls 33339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-0g 17069  df-gsum 17070  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-frmd 18403  df-mrex 33348  df-mex 33349  df-mrsub 33352  df-msub 33353  df-mvh 33354  df-mpst 33355  df-msr 33356  df-msta 33357  df-mfs 33358  df-mcls 33359

This theorem is referenced by:  mclsssv  33426  ssmclslem  33427  ss2mcls  33430  mclsax  33431  mclsind  33432