Theorem mclsax 35763

Description: The closure is closed under axiom application. (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	⊢ (𝜑 → 𝑇 ∈ mFS)
mclsval.2	⊢ (𝜑 → 𝐾 ⊆ 𝐷)
mclsval.3	⊢ (𝜑 → 𝐵 ⊆ 𝐸)
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)
mclsax.4	⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
mclsax.5	⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
mclsax.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
mclsax.7	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
mclsax.8	⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)

Assertion

Ref	Expression
mclsax	⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

Distinct variable groups:   𝑣,𝐸   𝑎,𝑏,𝑣,𝑥,𝐻   𝑦,𝑣,𝐵,𝑥   𝑣,𝐶,𝑥   𝑥,𝐿,𝑦   𝑥,𝑂,𝑦   𝑦,𝑎,𝑆,𝑏,𝑣,𝑥   𝑀,𝑎,𝑏,𝑥,𝑦   𝑥,𝑃,𝑦   𝑥,𝑇,𝑦   𝜑,𝑎,𝑏,𝑣,𝑥,𝑦   𝑣,𝑉,𝑥   𝑊,𝑎,𝑏,𝑥   𝐾,𝑎,𝑏,𝑣,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐶(𝑦,𝑎,𝑏)   𝐷(𝑥,𝑦,𝑣,𝑎,𝑏)   𝑃(𝑣,𝑎,𝑏)   𝑇(𝑣,𝑎,𝑏)   𝐸(𝑥,𝑦,𝑎,𝑏)   𝐻(𝑦)   𝐿(𝑣,𝑎,𝑏)   𝑀(𝑣)   𝑂(𝑣,𝑎,𝑏)   𝑉(𝑦,𝑎,𝑏)   𝑊(𝑦,𝑣)

Proof of Theorem mclsax

Dummy variables 𝑐 𝑚 𝑜 𝑝 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		abid 2718	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ↔ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
2		intss1 4918	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝑐)
3	1, 2	sylbir 235	. . . . . . 7 ⊢ (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝑐)
4		mclsval.d	. . . . . . . . 9 ⊢ 𝐷 = (mDV‘𝑇)
5		mclsval.e	. . . . . . . . 9 ⊢ 𝐸 = (mEx‘𝑇)
6		mclsval.c	. . . . . . . . 9 ⊢ 𝐶 = (mCls‘𝑇)
7		mclsval.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ mFS)
8		mclsval.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ⊆ 𝐷)
9		mclsval.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
10		mclsax.h	. . . . . . . . 9 ⊢ 𝐻 = (mVH‘𝑇)
11		mclsax.a	. . . . . . . . 9 ⊢ 𝐴 = (mAx‘𝑇)
12		mclsax.l	. . . . . . . . 9 ⊢ 𝐿 = (mSubst‘𝑇)
13		mclsax.w	. . . . . . . . 9 ⊢ 𝑊 = (mVars‘𝑇)
14	4, 5, 6, 7, 8, 9, 10, 11, 12, 13	mclsval 35757	. . . . . . . 8 ⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
15	14	sseq1d 3965	. . . . . . 7 ⊢ (𝜑 → ((𝐾𝐶𝐵) ⊆ 𝑐 ↔ ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝑐))
16	3, 15	imbitrrid 246	. . . . . 6 ⊢ (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝐾𝐶𝐵) ⊆ 𝑐))
17		sstr2 3940	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) → ((𝐾𝐶𝐵) ⊆ 𝑐 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
18	17	com12 32	. . . . . . . . . . . . . 14 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
19	18	anim1d 611	. . . . . . . . . . . . 13 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾))))
20	19	imim1d 82	. . . . . . . . . . . 12 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))
21	20	ralimdv 3150	. . . . . . . . . . 11 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))
22	21	imim2d 57	. . . . . . . . . 10 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
23	22	alimdv 1917	. . . . . . . . 9 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
24	23	2alimdv 1919	. . . . . . . 8 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
25	24	com12 32	. . . . . . 7 ⊢ (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → ((𝐾𝐶𝐵) ⊆ 𝑐 → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
26	25	adantl 481	. . . . . 6 ⊢ (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → ((𝐾𝐶𝐵) ⊆ 𝑐 → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
27	16, 26	sylcom 30	. . . . 5 ⊢ (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))))
28		eqid 2736	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
29		eqid 2736	. . . . . . . 8 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
30	28, 29	mstapst 35741	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
31	11, 29	maxsta 35748	. . . . . . . . 9 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
32	7, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ (mStat‘𝑇))
33		mclsax.4	. . . . . . . 8 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
34	32, 33	sseldd 3934	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
35	30, 34	sselid 3931	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
36	28	mpstrcl 35735	. . . . . 6 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) → (𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V))
37		simp1 1136	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → 𝑚 = 𝑀)
38		simp2 1137	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → 𝑜 = 𝑂)
39		simp3 1138	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
40	37, 38, 39	oteq123d 4844	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
41	40	eleq1d 2821	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴))
42	38	uneq1d 4119	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑜 ∪ ran 𝐻) = (𝑂 ∪ ran 𝐻))
43	42	imaeq2d 6019	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑠 “ (𝑜 ∪ ran 𝐻)) = (𝑠 “ (𝑂 ∪ ran 𝐻)))
44	43	sseq1d 3965	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
45	37	breqd 5109	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑥𝑚𝑦 ↔ 𝑥𝑀𝑦))
46	45	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)))
47	46	2albidv 1924	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) ↔ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)))
48	44, 47	anbi12d 632	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) ↔ ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾))))
49	39	fveq2d 6838	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑠‘𝑝) = (𝑠‘𝑃))
50	49	eleq1d 2821	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((𝑠‘𝑝) ∈ 𝑐 ↔ (𝑠‘𝑃) ∈ 𝑐))
51	48, 50	imbi12d 344	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐)))
52	51	ralbidv 3159	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐)))
53	41, 52	imbi12d 344	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐))))
54	53	spc3gv 3558	. . . . . 6 ⊢ ((𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V) → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐))))
55	35, 36, 54	3syl 18	. . . . 5 ⊢ (𝜑 → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐))))
56		elun 4105	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑂 ∪ ran 𝐻) ↔ (𝑥 ∈ 𝑂 ∨ 𝑥 ∈ ran 𝐻))
57		mclsax.6	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
58		mclsax.7	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
59	58	ralrimiva 3128	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
60		mclsax.v	. . . . . . . . . . . . . . . . 17 ⊢ 𝑉 = (mVR‘𝑇)
61	60, 5, 10	mvhf 35752	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
62	7, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
63		ffn 6662	. . . . . . . . . . . . . . 15 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
64		fveq2 6834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝐻‘𝑣) → (𝑆‘𝑥) = (𝑆‘(𝐻‘𝑣)))
65	64	eleq1d 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐻‘𝑣) → ((𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
66	65	ralrn 7033	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Fn 𝑉 → (∀𝑥 ∈ ran 𝐻(𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
67	62, 63, 66	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐻(𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
68	59, 67	mpbird 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐻(𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
69	68	r19.21bi 3228	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
70	57, 69	jaodan 959	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑂 ∨ 𝑥 ∈ ran 𝐻)) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
71	56, 70	sylan2b 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑂 ∪ ran 𝐻)) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
72	71	ralrimiva 3128	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
73		mclsax.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
74	12, 5	msubf 35726	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
75	73, 74	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
76	75	ffund 6666	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑆)
77	4, 5, 28	elmpst 35730	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
78	35, 77	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
79	78	simp2d 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin))
80	79	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ⊆ 𝐸)
81	75	fdmd 6672	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑆 = 𝐸)
82	80, 81	sseqtrrd 3971	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
83	62	frnd 6670	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
84	83, 81	sseqtrrd 3971	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐻 ⊆ dom 𝑆)
85	82, 84	unssd 4144	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆)
86		funimass4 6898	. . . . . . . . . 10 ⊢ ((Fun 𝑆 ∧ (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆) → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
87	76, 85, 86	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
88	72, 87	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵))
89		mclsax.8	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)
90	89	3exp2 1355	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥𝑀𝑦 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))) → 𝑎𝐾𝑏))))
91	90	imp4b 421	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))) → 𝑎𝐾𝑏))
92	91	ralrimivv 3177	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))𝑎𝐾𝑏)
93		dfss3 3922	. . . . . . . . . . . 12 ⊢ (((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾 ↔ ∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦))))𝑧 ∈ 𝐾)
94		eleq1 2824	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
95		df-br 5099	. . . . . . . . . . . . . 14 ⊢ (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
96	94, 95	bitr4di 289	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ 𝑎𝐾𝑏))
97	96	ralxp 5790	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦))))𝑧 ∈ 𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))𝑎𝐾𝑏)
98	93, 97	bitri 275	. . . . . . . . . . 11 ⊢ (((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾 ↔ ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))𝑎𝐾𝑏)
99	92, 98	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)
100	99	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
101	100	alrimivv 1929	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
102	88, 101	jca 511	. . . . . . 7 ⊢ (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)))
103		imaeq1 6014	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑠 “ (𝑂 ∪ ran 𝐻)) = (𝑆 “ (𝑂 ∪ ran 𝐻)))
104	103	sseq1d 3965	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
105		fveq1 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑥)))
106	105	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑥))))
107		fveq1 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑦)))
108	107	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑦))))
109	106, 108	xpeq12d 5655	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) = ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))))
110	109	sseq1d 3965	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾 ↔ ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
111	110	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → ((𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) ↔ (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)))
112	111	2albidv 1924	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾) ↔ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)))
113	104, 112	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) ↔ ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))))
114		fveq1 6833	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑃) = (𝑆‘𝑃))
115	114	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ((𝑠‘𝑃) ∈ 𝑐 ↔ (𝑆‘𝑃) ∈ 𝑐))
116	113, 115	imbi12d 344	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐) ↔ (((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑆‘𝑃) ∈ 𝑐)))
117	116	rspcv 3572	. . . . . . . 8 ⊢ (𝑆 ∈ ran 𝐿 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐) → (((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑆‘𝑃) ∈ 𝑐)))
118	73, 117	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐) → (((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑆‘𝑃) ∈ 𝑐)))
119	102, 118	mpid 44	. . . . . 6 ⊢ (𝜑 → (∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐) → (𝑆‘𝑃) ∈ 𝑐))
120	33, 119	embantd 59	. . . . 5 ⊢ (𝜑 → ((⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑃) ∈ 𝑐)) → (𝑆‘𝑃) ∈ 𝑐))
121	27, 55, 120	3syld 60	. . . 4 ⊢ (𝜑 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝑆‘𝑃) ∈ 𝑐))
122	121	alrimiv 1928	. . 3 ⊢ (𝜑 → ∀𝑐(((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝑆‘𝑃) ∈ 𝑐))
123		fvex 6847	. . . 4 ⊢ (𝑆‘𝑃) ∈ V
124	123	elintab 4914	. . 3 ⊢ ((𝑆‘𝑃) ∈ ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ↔ ∀𝑐(((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → (𝑆‘𝑃) ∈ 𝑐))
125	122, 124	sylibr 234	. 2 ⊢ (𝜑 → (𝑆‘𝑃) ∈ ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝐿(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
126	125, 14	eleqtrrd 2839	1 ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  Vcvv 3440   ∪ cun 3899   ⊆ wss 3901  ⟨cop 4586  ⟨cotp 4588  ∩ cint 4902   class class class wbr 5098   × cxp 5622  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Fincfn 8883  mVRcmvar 35655  mAxcmax 35659  mExcmex 35661  mDVcmdv 35662  mVarscmvrs 35663  mSubstcmsub 35665  mVHcmvh 35666  mPreStcmpst 35667  mStatcmsta 35669  mFScmfs 35670  mClscmcls 35671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-word 14437  df-concat 14494  df-s1 14520  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-0g 17361  df-gsum 17362  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-frmd 18774  df-mrex 35680  df-mex 35681  df-mrsub 35684  df-msub 35685  df-mvh 35686  df-mpst 35687  df-msr 35688  df-msta 35689  df-mfs 35690  df-mcls 35691

This theorem is referenced by:  mclsppslem  35777