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Theorem mclsind 34549
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 34542 . 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 4231 . . . . . 6 (𝜑𝐵 ⊆ (𝐸𝑄))
14 mclsax.v . . . . . . . . . . 11 𝑉 = (mVR‘𝑇)
1514, 2, 7mvhf 34537 . . . . . . . . . 10 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
164, 15syl 17 . . . . . . . . 9 (𝜑𝐻:𝑉𝐸)
1716ffnd 6715 . . . . . . . 8 (𝜑𝐻 Fn 𝑉)
1816ffvelcdmda 7083 . . . . . . . . . 10 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝐸)
19 mclsind.5 . . . . . . . . . 10 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝑄)
2018, 19elind 4193 . . . . . . . . 9 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ (𝐸𝑄))
2120ralrimiva 3146 . . . . . . . 8 (𝜑 → ∀𝑣𝑉 (𝐻𝑣) ∈ (𝐸𝑄))
22 ffnfv 7114 . . . . . . . 8 (𝐻:𝑉⟶(𝐸𝑄) ↔ (𝐻 Fn 𝑉 ∧ ∀𝑣𝑉 (𝐻𝑣) ∈ (𝐸𝑄)))
2317, 21, 22sylanbrc 583 . . . . . . 7 (𝜑𝐻:𝑉⟶(𝐸𝑄))
2423frnd 6722 . . . . . 6 (𝜑 → ran 𝐻 ⊆ (𝐸𝑄))
2513, 24unssd 4185 . . . . 5 (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄))
26 id 22 . . . . . . . . . . . 12 ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄))
27 inss2 4228 . . . . . . . . . . . 12 (𝐸𝑄) ⊆ 𝑄
2826, 27sstrdi 3993 . . . . . . . . . . 11 ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)
294adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑇 ∈ mFS)
30 eqid 2732 . . . . . . . . . . . . . . . . . . . . 21 (mREx‘𝑇) = (mREx‘𝑇)
3114, 30, 9, 2msubff 34509 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ mFS → 𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸m 𝐸))
32 frn 6721 . . . . . . . . . . . . . . . . . . . 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 3982 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ (𝐸m 𝐸))
36 elmapi 8839 . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . . . 22 (mStat‘𝑇) = (mStat‘𝑇)
398, 38maxsta 34533 . . . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . . 21 (mPreSt‘𝑇) = (mPreSt‘𝑇)
4241, 38mstapst 34526 . . . . . . . . . . . . . . . . . . . 20 (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
4340, 42sstrdi 3993 . . . . . . . . . . . . . . . . . . 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 3982 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
461, 2, 41elmpst 34515 . . . . . . . . . . . . . . . . . . 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 7084 . . . . . . . . . . . . . . . 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 4193 . . . . . . . . . . . . . 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 418 . . . . . . . . . . 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 411 . . . . . . . . 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 3146 . . . . . . . 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 413 . . . . . . 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 1930 . . . . . 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 1931 . . . . 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 6902 . . . . . . 7 𝐸 ∈ V
6362inex1 5316 . . . . . 6 (𝐸𝑄) ∈ V
64 sseq2 4007 . . . . . . 7 (𝑐 = (𝐸𝑄) → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄)))
65 sseq2 4007 . . . . . . . . . . . . 13 (𝑐 = (𝐸𝑄) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄)))
6665anbi1d 630 . . . . . . . . . . . 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 2822 . . . . . . . . . . . 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 1923 . . . . . . . 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 1926 . . . . . . 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 631 . . . . . 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 3667 . . . . 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 4966 . . . 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 3993 . 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 4019 1 (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  {cab 2709  wral 3061  cun 3945  cin 3946  wss 3947  cotp 4635   cint 4949   class class class wbr 5147   × cxp 5673  ccnv 5674  ran crn 5676  cima 5678   Fn wfn 6535  wf 6536  cfv 6540  (class class class)co 7405  m cmap 8816  pm cpm 8817  Fincfn 8935  mVRcmvar 34440  mAxcmax 34444  mRExcmrex 34445  mExcmex 34446  mDVcmdv 34447  mVarscmvrs 34448  mSubstcmsub 34450  mVHcmvh 34451  mPreStcmpst 34452  mStatcmsta 34454  mFScmfs 34455  mClscmcls 34456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-word 14461  df-concat 14517  df-s1 14542  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-0g 17383  df-gsum 17384  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-frmd 18726  df-mrex 34465  df-mex 34466  df-mrsub 34469  df-msub 34470  df-mvh 34471  df-mpst 34472  df-msr 34473  df-msta 34474  df-mfs 34475  df-mcls 34476
This theorem is referenced by:  mclspps  34563
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