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Theorem mclsval 32059
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 (𝜑𝑇 ∈ mFS)
mclsval.2 (𝜑𝐾𝐷)
mclsval.3 (𝜑𝐵𝐸)
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 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 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 𝑑 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mclsval.c . . 3 𝐶 = (mCls‘𝑇)
2 mclsval.1 . . . 4 (𝜑𝑇 ∈ mFS)
3 elex 3414 . . . 4 (𝑇 ∈ mFS → 𝑇 ∈ V)
4 fveq2 6446 . . . . . . . 8 (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
5 mclsval.d . . . . . . . 8 𝐷 = (mDV‘𝑇)
64, 5syl6eqr 2832 . . . . . . 7 (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
76pweqd 4384 . . . . . 6 (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝐷)
8 fveq2 6446 . . . . . . . 8 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
9 mclsval.e . . . . . . . 8 𝐸 = (mEx‘𝑇)
108, 9syl6eqr 2832 . . . . . . 7 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
1110pweqd 4384 . . . . . 6 (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
12 fveq2 6446 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (mVH‘𝑡) = (mVH‘𝑇))
13 mclsval.h . . . . . . . . . . . . 13 𝐻 = (mVH‘𝑇)
1412, 13syl6eqr 2832 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (mVH‘𝑡) = 𝐻)
1514rneqd 5598 . . . . . . . . . . 11 (𝑡 = 𝑇 → ran (mVH‘𝑡) = ran 𝐻)
1615uneq2d 3990 . . . . . . . . . 10 (𝑡 = 𝑇 → ( ∪ ran (mVH‘𝑡)) = ( ∪ ran 𝐻))
1716sseq1d 3851 . . . . . . . . 9 (𝑡 = 𝑇 → (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ↔ ( ∪ ran 𝐻) ⊆ 𝑐))
18 fveq2 6446 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
19 mclsval.a . . . . . . . . . . . . . 14 𝐴 = (mAx‘𝑇)
2018, 19syl6eqr 2832 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
2120eleq2d 2845 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) ↔ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴))
22 fveq2 6446 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (mSubst‘𝑡) = (mSubst‘𝑇))
23 mclsval.s . . . . . . . . . . . . . . 15 𝑆 = (mSubst‘𝑇)
2422, 23syl6eqr 2832 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (mSubst‘𝑡) = 𝑆)
2524rneqd 5598 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ran (mSubst‘𝑡) = ran 𝑆)
2615uneq2d 3990 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → (𝑜 ∪ ran (mVH‘𝑡)) = (𝑜 ∪ ran 𝐻))
2726imaeq2d 5720 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → (𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) = (𝑠 “ (𝑜 ∪ ran 𝐻)))
2827sseq1d 3851 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → ((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
29 fveq2 6446 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
30 mclsval.v . . . . . . . . . . . . . . . . . . . . 21 𝑉 = (mVars‘𝑇)
3129, 30syl6eqr 2832 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
3214fveq1d 6448 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → ((mVH‘𝑡)‘𝑥) = (𝐻𝑥))
3332fveq2d 6450 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → (𝑠‘((mVH‘𝑡)‘𝑥)) = (𝑠‘(𝐻𝑥)))
3431, 33fveq12d 6453 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) = (𝑉‘(𝑠‘(𝐻𝑥))))
3514fveq1d 6448 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → ((mVH‘𝑡)‘𝑦) = (𝐻𝑦))
3635fveq2d 6450 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → (𝑠‘((mVH‘𝑡)‘𝑦)) = (𝑠‘(𝐻𝑦)))
3731, 36fveq12d 6453 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦))) = (𝑉‘(𝑠‘(𝐻𝑦))))
3834, 37xpeq12d 5386 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) = ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))))
3938sseq1d 3851 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → ((((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑 ↔ ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑))
4039imbi2d 332 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇 → ((𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) ↔ (𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)))
41402albidv 1966 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇 → (∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) ↔ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)))
4228, 41anbi12d 624 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) ↔ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑))))
4342imbi1d 333 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))
4425, 43raleqbidv 3326 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))
4521, 44imbi12d 336 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)) ↔ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐))))
4645albidv 1963 . . . . . . . . . 10 (𝑡 = 𝑇 → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)) ↔ ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐))))
47462albidv 1966 . . . . . . . . 9 (𝑡 = 𝑇 → (∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)) ↔ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐))))
4817, 47anbi12d 624 . . . . . . . 8 (𝑡 = 𝑇 → ((( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐))) ↔ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))))
4948abbidv 2906 . . . . . . 7 (𝑡 = 𝑇 → {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))} = {𝑐 ∣ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))})
5049inteqd 4715 . . . . . 6 (𝑡 = 𝑇 {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))} = {𝑐 ∣ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))})
517, 11, 50mpt2eq123dv 6994 . . . . 5 (𝑡 = 𝑇 → (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}) = (𝑑 ∈ 𝒫 𝐷, ∈ 𝒫 𝐸 {𝑐 ∣ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}))
52 df-mcls 31993 . . . . 5 mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}))
535fvexi 6460 . . . . . . 7 𝐷 ∈ V
5453pwex 5092 . . . . . 6 𝒫 𝐷 ∈ V
559fvexi 6460 . . . . . . 7 𝐸 ∈ V
5655pwex 5092 . . . . . 6 𝒫 𝐸 ∈ V
5754, 56mpt2ex 7527 . . . . 5 (𝑑 ∈ 𝒫 𝐷, ∈ 𝒫 𝐸 {𝑐 ∣ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}) ∈ V
5851, 52, 57fvmpt 6542 . . . 4 (𝑇 ∈ V → (mCls‘𝑇) = (𝑑 ∈ 𝒫 𝐷, ∈ 𝒫 𝐸 {𝑐 ∣ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}))
592, 3, 583syl 18 . . 3 (𝜑 → (mCls‘𝑇) = (𝑑 ∈ 𝒫 𝐷, ∈ 𝒫 𝐸 {𝑐 ∣ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}))
601, 59syl5eq 2826 . 2 (𝜑𝐶 = (𝑑 ∈ 𝒫 𝐷, ∈ 𝒫 𝐸 {𝑐 ∣ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}))
61 simprr 763 . . . . . . 7 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → = 𝐵)
6261uneq1d 3989 . . . . . 6 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → ( ∪ ran 𝐻) = (𝐵 ∪ ran 𝐻))
6362sseq1d 3851 . . . . 5 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → (( ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ 𝑐))
64 simprl 761 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → 𝑑 = 𝐾)
6564sseq2d 3852 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → (((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑 ↔ ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾))
6665imbi2d 332 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → ((𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑) ↔ (𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)))
67662albidv 1966 . . . . . . . . . . 11 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → (∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑) ↔ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)))
6867anbi2d 622 . . . . . . . . . 10 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) ↔ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾))))
6968imbi1d 333 . . . . . . . . 9 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))
7069ralbidv 3168 . . . . . . . 8 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → (∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))
7170imbi2d 332 . . . . . . 7 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)) ↔ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
7271albidv 1963 . . . . . 6 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)) ↔ ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
73722albidv 1966 . . . . 5 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → (∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)) ↔ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))))
7463, 73anbi12d 624 . . . 4 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → ((( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐))) ↔ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))))
7574abbidv 2906 . . 3 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → {𝑐 ∣ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))} = {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))})
7675inteqd 4715 . 2 ((𝜑 ∧ (𝑑 = 𝐾 = 𝐵)) → {𝑐 ∣ (( ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))} = {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))})
77 mclsval.2 . . 3 (𝜑𝐾𝐷)
7853elpw2 5062 . . 3 (𝐾 ∈ 𝒫 𝐷𝐾𝐷)
7977, 78sylibr 226 . 2 (𝜑𝐾 ∈ 𝒫 𝐷)
80 mclsval.3 . . 3 (𝜑𝐵𝐸)
8155elpw2 5062 . . 3 (𝐵 ∈ 𝒫 𝐸𝐵𝐸)
8280, 81sylibr 226 . 2 (𝜑𝐵 ∈ 𝒫 𝐸)
835, 9, 1, 2, 77, 80, 13, 19, 23, 30mclsssvlem 32058 . . 3 (𝜑 {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ 𝐸)
8455ssex 5039 . . 3 ( {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ 𝐸 {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ∈ V)
8583, 84syl 17 . 2 (𝜑 {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ∈ V)
8660, 76, 79, 82, 85ovmpt2d 7065 1 (𝜑 → (𝐾𝐶𝐵) = {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wal 1599   = wceq 1601  wcel 2107  {cab 2763  wral 3090  Vcvv 3398  cun 3790  wss 3792  𝒫 cpw 4379  cotp 4406   cint 4710   class class class wbr 4886   × cxp 5353  ran crn 5356  cima 5358  cfv 6135  (class class class)co 6922  cmpt2 6924  mAxcmax 31961  mExcmex 31963  mDVcmdv 31964  mVarscmvrs 31965  mSubstcmsub 31967  mVHcmvh 31968  mFScmfs 31972  mClscmcls 31973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-ot 4407  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-seq 13120  df-hash 13436  df-word 13600  df-concat 13661  df-s1 13686  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-0g 16488  df-gsum 16489  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-submnd 17722  df-frmd 17773  df-mrex 31982  df-mex 31983  df-mrsub 31986  df-msub 31987  df-mvh 31988  df-mpst 31989  df-msr 31990  df-msta 31991  df-mfs 31992  df-mcls 31993
This theorem is referenced by:  mclsssv  32060  ssmclslem  32061  ss2mcls  32064  mclsax  32065  mclsind  32066
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