Theorem mclsval 35535

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 3459	. . . 4 ⊢ (𝑇 ∈ mFS → 𝑇 ∈ V)
4		fveq2 6826	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
5		mclsval.d	. . . . . . . 8 ⊢ 𝐷 = (mDV‘𝑇)
6	4, 5	eqtr4di 2782	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
7	6	pweqd 4570	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝐷)
8		fveq2 6826	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
9		mclsval.e	. . . . . . . 8 ⊢ 𝐸 = (mEx‘𝑇)
10	8, 9	eqtr4di 2782	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
11	10	pweqd 4570	. . . . . 6 ⊢ (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
12		fveq2 6826	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mVH‘𝑡) = (mVH‘𝑇))
13		mclsval.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (mVH‘𝑇)
14	12, 13	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (mVH‘𝑡) = 𝐻)
15	14	rneqd 5884	. . . . . . . . . . 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 3969	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ↔ (ℎ ∪ ran 𝐻) ⊆ 𝑐))
18		fveq2 6826	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
19		mclsval.a	. . . . . . . . . . . . . 14 ⊢ 𝐴 = (mAx‘𝑇)
20	18, 19	eqtr4di 2782	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
21	20	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) ↔ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴))
22		fveq2 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → (mSubst‘𝑡) = (mSubst‘𝑇))
23		mclsval.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (mSubst‘𝑇)
24	22, 23	eqtr4di 2782	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (mSubst‘𝑡) = 𝑆)
25	24	rneqd 5884	. . . . . . . . . . . . 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 6015	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → (𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) = (𝑠 “ (𝑜 ∪ ran 𝐻)))
28	27	sseq1d 3969	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → ((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
29		fveq2 6826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
30		mclsval.v	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑉 = (mVars‘𝑇)
31	29, 30	eqtr4di 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
32	14	fveq1d 6828	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → ((mVH‘𝑡)‘𝑥) = (𝐻‘𝑥))
33	32	fveq2d 6830	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (𝑠‘((mVH‘𝑡)‘𝑥)) = (𝑠‘(𝐻‘𝑥)))
34	31, 33	fveq12d 6833	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) = (𝑉‘(𝑠‘(𝐻‘𝑥))))
35	14	fveq1d 6828	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → ((mVH‘𝑡)‘𝑦) = (𝐻‘𝑦))
36	35	fveq2d 6830	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → (𝑠‘((mVH‘𝑡)‘𝑦)) = (𝑠‘(𝐻‘𝑦)))
37	31, 36	fveq12d 6833	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦))) = (𝑉‘(𝑠‘(𝐻‘𝑦))))
38	34, 37	xpeq12d 5654	. . . . . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . . . . 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 1923	. . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 3310	. . . . . . . . . . . 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 1920	. . . . . . . . . 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 1923	. . . . . . . . 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 632	. . . . . . . 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 2795	. . . . . . 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 4904	. . . . . 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 7428	. . . . 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 35469	. . . . 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 6840	. . . . . . 7 ⊢ 𝐷 ∈ V
54	53	pwex 5322	. . . . . 6 ⊢ 𝒫 𝐷 ∈ V
55	9	fvexi 6840	. . . . . . 7 ⊢ 𝐸 ∈ V
56	55	pwex 5322	. . . . . 6 ⊢ 𝒫 𝐸 ∈ V
57	54, 56	mpoex 8021	. . . . 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 6934	. . . 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	eqtrid 2776	. 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 772	. . . . . . 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 3969	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → ((ℎ ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ 𝑐))
64		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑑 = 𝐾 ∧ ℎ = 𝐵)) → 𝑑 = 𝐾)
65	64	sseq2d 3970	. . . . . . . . . . . . 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 1923	. . . . . . . . . . 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 630	. . . . . . . . . 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 3152	. . . . . . . 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 1920	. . . . . 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 1923	. . . . 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 632	. . . 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 2795	. . 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 4904	. 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 5276	. . 3 ⊢ (𝐾 ∈ 𝒫 𝐷 ↔ 𝐾 ⊆ 𝐷)
79	77, 78	sylibr 234	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝒫 𝐷)
80		mclsval.3	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
81	55	elpw2 5276	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐸 ↔ 𝐵 ⊆ 𝐸)
82	80, 81	sylibr 234	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐸)
83	5, 9, 1, 2, 77, 80, 13, 19, 23, 30	mclsssvlem 35534	. . 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 5263	. . 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 7505	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 1538   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  Vcvv 3438   ∪ cun 3903   ⊆ wss 3905  𝒫 cpw 4553  ⟨cotp 4587  ∩ cint 4899   class class class wbr 5095   × cxp 5621  ran crn 5624   “ cima 5626  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  mAxcmax 35437  mExcmex 35439  mDVcmdv 35440  mVarscmvrs 35441  mSubstcmsub 35443  mVHcmvh 35444  mFScmfs 35448  mClscmcls 35449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-seq 13927  df-hash 14256  df-word 14439  df-concat 14496  df-s1 14521  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  df-gsum 17364  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-submnd 18676  df-frmd 18741  df-mrex 35458  df-mex 35459  df-mrsub 35462  df-msub 35463  df-mvh 35464  df-mpst 35465  df-msr 35466  df-msta 35467  df-mfs 35468  df-mcls 35469

This theorem is referenced by:  mclsssv  35536  ssmclslem  35537  ss2mcls  35540  mclsax  35541  mclsind  35542