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Theorem mclsind 31593
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 31586 . 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 3870 . . . . . 6 (𝜑𝐵 ⊆ (𝐸𝑄))
14 mclsax.v . . . . . . . . . . 11 𝑉 = (mVR‘𝑇)
1514, 2, 7mvhf 31581 . . . . . . . . . 10 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
164, 15syl 17 . . . . . . . . 9 (𝜑𝐻:𝑉𝐸)
17 ffn 6083 . . . . . . . . 9 (𝐻:𝑉𝐸𝐻 Fn 𝑉)
1816, 17syl 17 . . . . . . . 8 (𝜑𝐻 Fn 𝑉)
1916ffvelrnda 6399 . . . . . . . . . 10 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝐸)
20 mclsind.5 . . . . . . . . . 10 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝑄)
2119, 20elind 3831 . . . . . . . . 9 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ (𝐸𝑄))
2221ralrimiva 2995 . . . . . . . 8 (𝜑 → ∀𝑣𝑉 (𝐻𝑣) ∈ (𝐸𝑄))
23 ffnfv 6428 . . . . . . . 8 (𝐻:𝑉⟶(𝐸𝑄) ↔ (𝐻 Fn 𝑉 ∧ ∀𝑣𝑉 (𝐻𝑣) ∈ (𝐸𝑄)))
2418, 22, 23sylanbrc 699 . . . . . . 7 (𝜑𝐻:𝑉⟶(𝐸𝑄))
25 frn 6091 . . . . . . 7 (𝐻:𝑉⟶(𝐸𝑄) → ran 𝐻 ⊆ (𝐸𝑄))
2624, 25syl 17 . . . . . 6 (𝜑 → ran 𝐻 ⊆ (𝐸𝑄))
2713, 26unssd 3822 . . . . 5 (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄))
28 id 22 . . . . . . . . . . . 12 ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄))
29 inss2 3867 . . . . . . . . . . . 12 (𝐸𝑄) ⊆ 𝑄
3028, 29syl6ss 3648 . . . . . . . . . . 11 ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)
314adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑇 ∈ mFS)
32 eqid 2651 . . . . . . . . . . . . . . . . . . . . 21 (mREx‘𝑇) = (mREx‘𝑇)
3314, 32, 9, 2msubff 31553 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ mFS → 𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸𝑚 𝐸))
34 frn 6091 . . . . . . . . . . . . . . . . . . . 20 (𝐿:((mREx‘𝑇) ↑pm 𝑉)⟶(𝐸𝑚 𝐸) → ran 𝐿 ⊆ (𝐸𝑚 𝐸))
3531, 33, 343syl 18 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ran 𝐿 ⊆ (𝐸𝑚 𝐸))
36 simpr2 1088 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ ran 𝐿)
3735, 36sseldd 3637 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠 ∈ (𝐸𝑚 𝐸))
38 elmapi 7921 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ (𝐸𝑚 𝐸) → 𝑠:𝐸𝐸)
3937, 38syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑠:𝐸𝐸)
40 eqid 2651 . . . . . . . . . . . . . . . . . . . . . 22 (mStat‘𝑇) = (mStat‘𝑇)
418, 40maxsta 31577 . . . . . . . . . . . . . . . . . . . . 21 (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
4231, 41syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mStat‘𝑇))
43 eqid 2651 . . . . . . . . . . . . . . . . . . . . 21 (mPreSt‘𝑇) = (mPreSt‘𝑇)
4443, 40mstapst 31570 . . . . . . . . . . . . . . . . . . . 20 (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
4542, 44syl6ss 3648 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝐴 ⊆ (mPreSt‘𝑇))
46 simpr1 1087 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴)
4745, 46sseldd 3637 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
481, 2, 43elmpst 31559 . . . . . . . . . . . . . . . . . . 19 (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚𝐷𝑚 = 𝑚) ∧ (𝑜𝐸𝑜 ∈ Fin) ∧ 𝑝𝐸))
4948simp3bi 1098 . . . . . . . . . . . . . . . . . 18 (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) → 𝑝𝐸)
5047, 49syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → 𝑝𝐸)
5139, 50ffvelrnd 6400 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄)) → (𝑠𝑝) ∈ 𝐸)
52513adant3 1101 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝐸)
53 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)
5452, 53elind 3831 . . . . . . . . . . . . . 14 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ (𝐸𝑄))
55543exp 1283 . . . . . . . . . . . . 13 (𝜑 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) → (∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾) → (𝑠𝑝) ∈ (𝐸𝑄))))
56553expd 1306 . . . . . . . . . . . 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))))))
5756imp31 447 . . . . . . . . . . 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))))
5830, 57syl5 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))))
5958impd 446 . . . . . . . . 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)))
6059ralrimiva 2995 . . . . . . . 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)))
6160ex 449 . . . . . . 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))))
6261alrimiv 1895 . . . . . 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))))
6362alrimivv 1896 . . . . 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))))
64 fvex 6239 . . . . . . . 8 (mEx‘𝑇) ∈ V
652, 64eqeltri 2726 . . . . . . 7 𝐸 ∈ V
6665inex1 4832 . . . . . 6 (𝐸𝑄) ∈ V
67 sseq2 3660 . . . . . . 7 (𝑐 = (𝐸𝑄) → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ (𝐸𝑄)))
68 sseq2 3660 . . . . . . . . . . . . 13 (𝑐 = (𝐸𝑄) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐸𝑄)))
6968anbi1d 741 . . . . . . . . . . . 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))))
70 eleq2 2719 . . . . . . . . . . . 12 (𝑐 = (𝐸𝑄) → ((𝑠𝑝) ∈ 𝑐 ↔ (𝑠𝑝) ∈ (𝐸𝑄)))
7169, 70imbi12d 333 . . . . . . . . . . 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))))
7271ralbidv 3015 . . . . . . . . . 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))))
7372imbi2d 329 . . . . . . . . 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)))))
7473albidv 1889 . . . . . . . 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)))))
75742albidv 1891 . . . . . . 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)))))
7667, 75anbi12d 747 . . . . . 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))))))
7766, 76elab 3382 . . . . 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)))))
7827, 63, 77sylanbrc 699 . . . 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)))})
79 intss1 4524 . . . 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))
8078, 79syl 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))
8180, 29syl6ss 3648 . 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)
8211, 81eqsstrd 3672 1 (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  cun 3605  cin 3606  wss 3607  cotp 4218   cint 4507   class class class wbr 4685   × cxp 5141  ccnv 5142  ran crn 5144  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  pm cpm 7900  Fincfn 7997  mVRcmvar 31484  mAxcmax 31488  mRExcmrex 31489  mExcmex 31490  mDVcmdv 31491  mVarscmvrs 31492  mSubstcmsub 31494  mVHcmvh 31495  mPreStcmpst 31496  mStatcmsta 31498  mFScmfs 31499  mClscmcls 31500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-frmd 17433  df-mrex 31509  df-mex 31510  df-mrsub 31513  df-msub 31514  df-mvh 31515  df-mpst 31516  df-msr 31517  df-msta 31518  df-mfs 31519  df-mcls 31520
This theorem is referenced by:  mclspps  31607
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