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Theorem mclsind 32805
 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 (𝜑𝑇 ∈ mFS)
mclsval.2 (𝜑𝐾𝐷)
mclsval.3 (𝜑𝐵𝐸)
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 (𝜑𝐵𝑄)
mclsind.5 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝑄)
mclsind.6 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 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.
StepHypRef 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 (𝜑𝑇 ∈ mFS)
5 mclsval.2 . . 3 (𝜑𝐾𝐷)
6 mclsval.3 . . 3 (𝜑𝐵𝐸)
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)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10mclsval 32798 . 2 (𝜑 → (𝐾𝐶𝐵) = {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))})
12 mclsind.4 . . . . . . 7 (𝜑𝐵𝑄)
136, 12ssind 4207 . . . . . 6 (𝜑𝐵 ⊆ (𝐸𝑄))
14 mclsax.v . . . . . . . . . . 11 𝑉 = (mVR‘𝑇)
1514, 2, 7mvhf 32793 . . . . . . . . . 10 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
164, 15syl 17 . . . . . . . . 9 (𝜑𝐻:𝑉𝐸)
1716ffnd 6508 . . . . . . . 8 (𝜑𝐻 Fn 𝑉)
1816ffvelrnda 6844 . . . . . . . . . 10 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝐸)
19 mclsind.5 . . . . . . . . . 10 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝑄)
2018, 19elind 4169 . . . . . . . . 9 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ (𝐸𝑄))
2120ralrimiva 3180 . . . . . . . 8 (𝜑 → ∀𝑣𝑉 (𝐻𝑣) ∈ (𝐸𝑄))
22 ffnfv 6875 . . . . . . . 8 (𝐻:𝑉⟶(𝐸𝑄) ↔ (𝐻 Fn 𝑉 ∧ ∀𝑣𝑉 (𝐻𝑣) ∈ (𝐸𝑄)))
2317, 21, 22sylanbrc 585 . . . . . . 7 (𝜑𝐻:𝑉⟶(𝐸𝑄))
2423frnd 6514 . . . . . 6 (𝜑 → ran 𝐻 ⊆ (𝐸𝑄))
2513, 24unssd 4160 . . . . 5 (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄))
26 id 22 . . . . . . . . . . . 12 ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄))
27 inss2 4204 . . . . . . . . . . . 12 (𝐸𝑄) ⊆ 𝑄
2826, 27sstrdi 3977 . . . . . . . . . . 11 ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)
294adantr 483 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑇 ∈ mFS)
30 eqid 2819 . . . . . . . . . . . . . . . . . . . . 21 (mREx‘𝑇) = (mREx‘𝑇)
3114, 30, 9, 2msubff 32765 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ mFS → 𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸m 𝐸))
32 frn 6513 . . . . . . . . . . . . . . . . . . . 20 (𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸m 𝐸) → ran 𝐿 ⊆ (𝐸m 𝐸))
3329, 31, 323syl 18 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ran 𝐿 ⊆ (𝐸m 𝐸))
34 simpr2 1189 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ ran 𝐿)
3533, 34sseldd 3966 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ (𝐸m 𝐸))
36 elmapi 8420 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ (𝐸m 𝐸) → 𝑠:𝐸𝐸)
3735, 36syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠:𝐸𝐸)
38 eqid 2819 . . . . . . . . . . . . . . . . . . . . . 22 (mStat‘𝑇) = (mStat‘𝑇)
398, 38maxsta 32789 . . . . . . . . . . . . . . . . . . . . 21 (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
4029, 39syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mStat‘𝑇))
41 eqid 2819 . . . . . . . . . . . . . . . . . . . . 21 (mPreSt‘𝑇) = (mPreSt‘𝑇)
4241, 38mstapst 32782 . . . . . . . . . . . . . . . . . . . 20 (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
4340, 42sstrdi 3977 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mPreSt‘𝑇))
44 simpr1 1188 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴)
4543, 44sseldd 3966 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
461, 2, 41elmpst 32771 . . . . . . . . . . . . . . . . . . 19 (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚𝐷𝑚 = 𝑚) ∧ (𝑜𝐸𝑜 ∈ Fin) ∧ 𝑝𝐸))
4746simp3bi 1141 . . . . . . . . . . . . . . . . . 18 (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) → 𝑝𝐸)
4845, 47syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑝𝐸)
4937, 48ffvelrnd 6845 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → (𝑠𝑝) ∈ 𝐸)
50493adant3 1126 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝐸)
51 mclsind.6 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑄)
5250, 51elind 4169 . . . . . . . . . . . . . 14 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄))
53523exp 1113 . . . . . . . . . . . . 13 (𝜑 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) → (∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) → (𝑠𝑝) ∈ (𝐸𝑄))))
54533expd 1347 . . . . . . . . . . . 12 (𝜑 → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → (𝑠 ∈ ran 𝐿 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄 → (∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) → (𝑠𝑝) ∈ (𝐸𝑄))))))
5554imp31 420 . . . . . . . . . . 11 (((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) ∧ 𝑠 ∈ ran 𝐿) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄 → (∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) → (𝑠𝑝) ∈ (𝐸𝑄))))
5628, 55syl5 34 . . . . . . . . . 10 (((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) ∧ 𝑠 ∈ ran 𝐿) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) → (∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) → (𝑠𝑝) ∈ (𝐸𝑄))))
5756impd 413 . . . . . . . . 9 (((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) ∧ 𝑠 ∈ ran 𝐿) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄)))
5857ralrimiva 3180 . . . . . . . 8 ((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄)))
5958ex 415 . . . . . . 7 (𝜑 → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄))))
6059alrimiv 1921 . . . . . 6 (𝜑 → ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄))))
6160alrimivv 1922 . . . . 5 (𝜑 → ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄))))
622fvexi 6677 . . . . . . 7 𝐸 ∈ V
6362inex1 5212 . . . . . 6 (𝐸𝑄) ∈ V
64 sseq2 3991 . . . . . . 7 (𝑐 = (𝐸𝑄) → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄)))
65 sseq2 3991 . . . . . . . . . . . . 13 (𝑐 = (𝐸𝑄) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄)))
6665anbi1d 631 . . . . . . . . . . . 12 (𝑐 = (𝐸𝑄) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) ↔ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾))))
67 eleq2 2899 . . . . . . . . . . . 12 (𝑐 = (𝐸𝑄) → ((𝑠𝑝) ∈ 𝑐 ↔ (𝑠𝑝) ∈ (𝐸𝑄)))
6866, 67imbi12d 347 . . . . . . . . . . 11 (𝑐 = (𝐸𝑄) → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄))))
6968ralbidv 3195 . . . . . . . . . 10 (𝑐 = (𝐸𝑄) → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄))))
7069imbi2d 343 . . . . . . . . 9 (𝑐 = (𝐸𝑄) → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) ↔ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄)))))
7170albidv 1914 . . . . . . . 8 (𝑐 = (𝐸𝑄) → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) ↔ ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄)))))
72712albidv 1917 . . . . . . 7 (𝑐 = (𝐸𝑄) → (∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)) ↔ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄)))))
7364, 72anbi12d 632 . . . . . 6 (𝑐 = (𝐸𝑄) → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐))) ↔ ((𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄) ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄))))))
7463, 73elab 3665 . . . . 5 ((𝐸𝑄) ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ↔ ((𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄) ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄)))))
7525, 61, 74sylanbrc 585 . . . 4 (𝜑 → (𝐸𝑄) ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))})
76 intss1 4882 . . . 4 ((𝐸𝑄) ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} → {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ (𝐸𝑄))
7775, 76syl 17 . . 3 (𝜑 {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ (𝐸𝑄))
7877, 27sstrdi 3977 . 2 (𝜑 {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ 𝑄)
7911, 78eqsstrd 4003 1 (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1081  ∀wal 1528   = wceq 1530   ∈ wcel 2107  {cab 2797  ∀wral 3136   ∪ cun 3932   ∩ cin 3933   ⊆ wss 3934  ⟨cotp 4567  ∩ cint 4867   class class class wbr 5057   × cxp 5546  ◡ccnv 5547  ran crn 5549   “ cima 5551   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148   ↑m cmap 8398   ↑pm cpm 8399  Fincfn 8501  mVRcmvar 32696  mAxcmax 32700  mRExcmrex 32701  mExcmex 32702  mDVcmdv 32703  mVarscmvrs 32704  mSubstcmsub 32706  mVHcmvh 32707  mPreStcmpst 32708  mStatcmsta 32710  mFScmfs 32711  mClscmcls 32712 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-ot 4568  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-seq 13362  df-hash 13683  df-word 13854  df-concat 13915  df-s1 13942  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-frmd 18006  df-mrex 32721  df-mex 32722  df-mrsub 32725  df-msub 32726  df-mvh 32727  df-mpst 32728  df-msr 32729  df-msta 32730  df-mfs 32731  df-mcls 32732 This theorem is referenced by:  mclspps  32819
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