Theorem mclsval 36021

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 3474	. . . 4 ⊢ (𝑇 ∈ mFS → 𝑇 ∈ V)
4		fveq2 6881	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
5		mclsval.d	. . . . . . . 8 ⊢ 𝐷 = (mDV‘𝑇)
6	4, 5	eqtr4di 2814	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
7	6	pweqd 4578	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝐷)
8		fveq2 6881	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
9		mclsval.e	. . . . . . . 8 ⊢ 𝐸 = (mEx‘𝑇)
10	8, 9	eqtr4di 2814	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
11	10	pweqd 4578	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
12		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mVH‘𝑡) = (mVH‘𝑇))
13		mclsval.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (mVH‘𝑇)
14	12, 13	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (mVH‘𝑡) = 𝐻)
15	14	rneqd 5928	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ran (mVH‘𝑡) = ran 𝐻)
16	15	uneq2d 4121	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (ℎ ∪ ran (mVH‘𝑡)) = (ℎ ∪ ran 𝐻))
17	16	sseq1d 3967	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ↔ (ℎ ∪ ran 𝐻) ⊆ 𝑐))
18		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
19		mclsval.a	. . . . . . . . . . . . . 14 ⊢ 𝐴 = (mAx‘𝑇)
20	18, 19	eqtr4di 2814	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
21	20	eleq2d 2847	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) ↔ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴))
22		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (mSubst‘𝑡) = (mSubst‘𝑇))
23		mclsval.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (mSubst‘𝑇)
24	22, 23	eqtr4di 2814	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (mSubst‘𝑡) = 𝑆)
25	24	rneqd 5928	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → ran (mSubst‘𝑡) = ran 𝑆)
26	15	uneq2d 4121	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → (𝑜 ∪ ran (mVH‘𝑡)) = (𝑜 ∪ ran 𝐻))
27	26	imaeq2d 6062	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → (𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) = (𝑠 “ (𝑜 ∪ ran 𝐻)))
28	27	sseq1d 3967	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → ((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
29		fveq2 6881	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
30		mclsval.v	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑉 = (mVars‘𝑇)
31	29, 30	eqtr4di 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
32	14	fveq1d 6883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → ((mVH‘𝑡)‘𝑥) = (𝐻‘𝑥))
33	32	fveq2d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (𝑠‘((mVH‘𝑡)‘𝑥)) = (𝑠‘(𝐻‘𝑥)))
34	31, 33	fveq12d 6888	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) = (𝑉‘(𝑠‘(𝐻‘𝑥))))
35	14	fveq1d 6883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → ((mVH‘𝑡)‘𝑦) = (𝐻‘𝑦))
36	35	fveq2d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (𝑠‘((mVH‘𝑡)‘𝑦)) = (𝑠‘(𝐻‘𝑦)))
37	31, 36	fveq12d 6888	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦))) = (𝑉‘(𝑠‘(𝐻‘𝑦))))
38	34, 37	xpeq12d 5692	. . . . . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . . . 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 1951	. . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . 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 344	. . . . . . . . . . . . 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 3336	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 1948	. . . . . . . . . 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 1951	. . . . . . . . 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 643	. . . . . . . 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 2827	. . . . . . 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 4916	. . . . . 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 7485	. . . . 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 35955	. . . . 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 6895	. . . . . . 7 ⊢ 𝐷 ∈ V
54	53	pwex 5351	. . . . . 6 ⊢ 𝒫 𝐷 ∈ V
55	9	fvexi 6895	. . . . . . 7 ⊢ 𝐸 ∈ V
56	55	pwex 5351	. . . . . 6 ⊢ 𝒫 𝐸 ∈ V
57	54, 56	mpoex 8075	. . . . 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 6989	. . . 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 19	. . 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	eqtrid 2808	. 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 784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → ℎ = 𝐵)
62	61	uneq1d 4120	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → (ℎ ∪ ran 𝐻) = (𝐵 ∪ ran 𝐻))
63	62	sseq1d 3967	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ 𝑐))
64		simprl 782	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → 𝑑 = 𝐾)
65	64	sseq2d 3968	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 1951	. . . . . . . . . . 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 641	. . . . . . . . . 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 344	. . . . . . . . 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 3186	. . . . . . . 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 343	. . . . . . 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 1948	. . . . . 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 1951	. . . . 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 643	. . . 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 2827	. . 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 4916	. 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 5304	. . 3 ⊢ (𝐾 ∈ 𝒫 𝐷 ↔ 𝐾 ⊆ 𝐷)
79	77, 78	sylibr 237	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝒫 𝐷)
80		mclsval.3	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
81	55	elpw2 5304	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐸 ↔ 𝐵 ⊆ 𝐸)
82	80, 81	sylibr 237	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐸)
83	5, 9, 1, 2, 77, 80, 13, 19, 23, 30	mclsssvlem 36020	. . 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 5290	. . 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 18	. 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 7562	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 400  ∀wal 1566   = wceq 1568   ∈ wcel 2141  {cab 2739  ∀wral 3077  Vcvv 3453   ∪ cun 3902   ⊆ wss 3904  𝒫 cpw 4561  ⟨cotp 4596  ∩ cint 4911   class class class wbr 5108   × cxp 5659  ran crn 5662   “ cima 5664  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  mAxcmax 35923  mExcmex 35925  mDVcmdv 35926  mVarscmvrs 35927  mSubstcmsub 35929  mVHcmvh 35930  mFScmfs 35934  mClscmcls 35935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-n0 12504  df-z 12591  df-uz 12862  df-fz 13535  df-fzo 13683  df-seq 14038  df-hash 14367  df-word 14551  df-concat 14608  df-s1 14634  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-0g 17493  df-gsum 17494  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-submnd 18841  df-frmd 18907  df-mrex 35944  df-mex 35945  df-mrsub 35948  df-msub 35949  df-mvh 35950  df-mpst 35951  df-msr 35952  df-msta 35953  df-mfs 35954  df-mcls 35955

This theorem is referenced by:  mclsssv  36022  ssmclslem  36023  ss2mcls  36026  mclsax  36027  mclsind  36028