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Theorem mclsind 35576
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 35569 . 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 4240 . . . . . 6 (𝜑𝐵 ⊆ (𝐸𝑄))
14 mclsax.v . . . . . . . . . . 11 𝑉 = (mVR‘𝑇)
1514, 2, 7mvhf 35564 . . . . . . . . . 10 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
164, 15syl 17 . . . . . . . . 9 (𝜑𝐻:𝑉𝐸)
1716ffnd 6736 . . . . . . . 8 (𝜑𝐻 Fn 𝑉)
1816ffvelcdmda 7103 . . . . . . . . . 10 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝐸)
19 mclsind.5 . . . . . . . . . 10 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝑄)
2018, 19elind 4199 . . . . . . . . 9 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ (𝐸𝑄))
2120ralrimiva 3145 . . . . . . . 8 (𝜑 → ∀𝑣𝑉 (𝐻𝑣) ∈ (𝐸𝑄))
22 ffnfv 7138 . . . . . . . 8 (𝐻:𝑉⟶(𝐸𝑄) ↔ (𝐻 Fn 𝑉 ∧ ∀𝑣𝑉 (𝐻𝑣) ∈ (𝐸𝑄)))
2317, 21, 22sylanbrc 583 . . . . . . 7 (𝜑𝐻:𝑉⟶(𝐸𝑄))
2423frnd 6743 . . . . . 6 (𝜑 → ran 𝐻 ⊆ (𝐸𝑄))
2513, 24unssd 4191 . . . . 5 (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄))
26 id 22 . . . . . . . . . . . 12 ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄))
27 inss2 4237 . . . . . . . . . . . 12 (𝐸𝑄) ⊆ 𝑄
2826, 27sstrdi 3995 . . . . . . . . . . 11 ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)
294adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑇 ∈ mFS)
30 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (mREx‘𝑇) = (mREx‘𝑇)
3114, 30, 9, 2msubff 35536 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ mFS → 𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸m 𝐸))
32 frn 6742 . . . . . . . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ ran 𝐿)
3533, 34sseldd 3983 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ (𝐸m 𝐸))
36 elmapi 8890 . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . . . 22 (mStat‘𝑇) = (mStat‘𝑇)
398, 38maxsta 35560 . . . . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . . 21 (mPreSt‘𝑇) = (mPreSt‘𝑇)
4241, 38mstapst 35553 . . . . . . . . . . . . . . . . . . . 20 (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
4340, 42sstrdi 3995 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mPreSt‘𝑇))
44 simpr1 1194 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴)
4543, 44sseldd 3983 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
461, 2, 41elmpst 35542 . . . . . . . . . . . . . . . . . . 19 (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚𝐷𝑚 = 𝑚) ∧ (𝑜𝐸𝑜 ∈ Fin) ∧ 𝑝𝐸))
4746simp3bi 1147 . . . . . . . . . . . . . . . . . 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, 48ffvelcdmd 7104 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → (𝑠𝑝) ∈ 𝐸)
50493adant3 1132 . . . . . . . . . . . . . . 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 4199 . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . 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 1353 . . . . . . . . . . . 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 417 . . . . . . . . . . 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 410 . . . . . . . . 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 3145 . . . . . . . 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 412 . . . . . . 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 1926 . . . . . 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 1927 . . . . 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 6919 . . . . . . 7 𝐸 ∈ V
6362inex1 5316 . . . . . 6 (𝐸𝑄) ∈ V
64 sseq2 4009 . . . . . . 7 (𝑐 = (𝐸𝑄) → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄)))
65 sseq2 4009 . . . . . . . . . . . . 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 2829 . . . . . . . . . . . 12 (𝑐 = (𝐸𝑄) → ((𝑠𝑝) ∈ 𝑐 ↔ (𝑠𝑝) ∈ (𝐸𝑄)))
6866, 67imbi12d 344 . . . . . . . . . . 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 3177 . . . . . . . . . 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 340 . . . . . . . . 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 1919 . . . . . . . 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 1922 . . . . . . 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 3678 . . . . 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 583 . . . 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 4962 . . . 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 3995 . 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 4017 1 (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wal 1537   = wceq 1539  wcel 2107  {cab 2713  wral 3060  cun 3948  cin 3949  wss 3950  cotp 4633   cint 4945   class class class wbr 5142   × cxp 5682  ccnv 5683  ran crn 5685  cima 5687   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  m cmap 8867  pm cpm 8868  Fincfn 8986  mVRcmvar 35467  mAxcmax 35471  mRExcmrex 35472  mExcmex 35473  mDVcmdv 35474  mVarscmvrs 35475  mSubstcmsub 35477  mVHcmvh 35478  mPreStcmpst 35479  mStatcmsta 35481  mFScmfs 35482  mClscmcls 35483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-word 14554  df-concat 14610  df-s1 14635  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-0g 17487  df-gsum 17488  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-submnd 18798  df-frmd 18863  df-mrex 35492  df-mex 35493  df-mrsub 35496  df-msub 35497  df-mvh 35498  df-mpst 35499  df-msr 35500  df-msta 35501  df-mfs 35502  df-mcls 35503
This theorem is referenced by:  mclspps  35590
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