Theorem mclsax 36039

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 2745	. . . . . . . 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 4929	. . . . . . . 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 238	. . . . . . 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 36033	. . . . . . . 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 3969	. . . . . . 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 249	. . . . . 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 3945	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) → ((𝐾𝐶𝐵) ⊆ 𝑐 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
18	17	com12 33	. . . . . . . . . . . . . 14 ⊢ ((𝐾𝐶𝐵) ⊆ 𝑐 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐))
19	18	anim1d 622	. . . . . . . . . . . . 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 83	. . . . . . . . . . . 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 3179	. . . . . . . . . . 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 58	. . . . . . . . . 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 1946	. . . . . . . . 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 1948	. . . . . . . 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 33	. . . . . . 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 486	. . . . . 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 31	. . . . 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 2763	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
29		eqid 2763	. . . . . . . 8 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
30	28, 29	mstapst 36017	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
31	11, 29	maxsta 36024	. . . . . . . . 9 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
32	7, 31	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ (mStat‘𝑇))
33		mclsax.4	. . . . . . . 8 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)
34	32, 33	sseldd 3939	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
35	30, 34	sselid 3936	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
36	28	mpstrcl 36011	. . . . . 6 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) → (𝑀 ∈ V ∧ 𝑂 ∈ V ∧ 𝑃 ∈ V))
37		simp1 1154	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → 𝑚 = 𝑀)
38		simp2 1155	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → 𝑜 = 𝑂)
39		simp3 1156	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
40	37, 38, 39	oteq123d 4854	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
41	40	eleq1d 2848	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴))
42	38	uneq1d 4122	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑜 ∪ ran 𝐻) = (𝑂 ∪ ran 𝐻))
43	42	imaeq2d 6064	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑠 “ (𝑜 ∪ ran 𝐻)) = (𝑠 “ (𝑂 ∪ ran 𝐻)))
44	43	sseq1d 3969	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
45	37	breqd 5121	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑥𝑚𝑦 ↔ 𝑥𝑀𝑦))
46	45	imbi1d 344	. . . . . . . . . . . 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 1953	. . . . . . . . . . 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 643	. . . . . . . . . 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 6887	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑠‘𝑝) = (𝑠‘𝑃))
50	49	eleq1d 2848	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((𝑠‘𝑝) ∈ 𝑐 ↔ (𝑠‘𝑃) ∈ 𝑐))
51	48, 50	imbi12d 347	. . . . . . . . 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 3188	. . . . . . . 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 347	. . . . . . 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 3564	. . . . . 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 19	. . . . 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 4108	. . . . . . . . . . 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 3157	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
60		mclsax.v	. . . . . . . . . . . . . . . . 17 ⊢ 𝑉 = (mVR‘𝑇)
61	60, 5, 10	mvhf 36028	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
62	7, 61	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
63		ffn 6707	. . . . . . . . . . . . . . 15 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
64		fveq2 6883	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝐻‘𝑣) → (𝑆‘𝑥) = (𝑆‘(𝐻‘𝑣)))
65	64	eleq1d 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐻‘𝑣) → ((𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
66	65	ralrn 7085	. . . . . . . . . . . . . . 15 ⊢ (𝐻 Fn 𝑉 → (∀𝑥 ∈ ran 𝐻(𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
67	62, 63, 66	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐻(𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
68	59, 67	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐻(𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
69	68	r19.21bi 3257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
70	57, 69	jaodan 972	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑂 ∨ 𝑥 ∈ ran 𝐻)) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
71	56, 70	sylan2b 605	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑂 ∪ ran 𝐻)) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
72	71	ralrimiva 3157	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
73		mclsax.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
74	12, 5	msubf 36002	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
75	73, 74	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
76	75	ffund 6712	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑆)
77	4, 5, 28	elmpst 36006	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
78	35, 77	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
79	78	simp2d 1161	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin))
80	79	simpld 499	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ⊆ 𝐸)
81	75	fdmd 6718	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑆 = 𝐸)
82	80, 81	sseqtrrd 3975	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
83	62	frnd 6716	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
84	83, 81	sseqtrrd 3975	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐻 ⊆ dom 𝑆)
85	82, 84	unssd 4146	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆)
86		funimass4 6947	. . . . . . . . . 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 595	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
88	72, 87	mpbird 260	. . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥𝑀𝑦 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))) → 𝑎𝐾𝑏))))
91	90	imp4b 426	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))) → 𝑎𝐾𝑏))
92	91	ralrimivv 3206	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))𝑎𝐾𝑏)
93		dfss3 3927	. . . . . . . . . . . 12 ⊢ (((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾 ↔ ∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦))))𝑧 ∈ 𝐾)
94		eleq1 2851	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
95		df-br 5111	. . . . . . . . . . . . . 14 ⊢ (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
96	94, 95	bitr4di 292	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ 𝑎𝐾𝑏))
97	96	ralxp 5829	. . . . . . . . . . . 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 278	. . . . . . . . . . 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 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)
100	99	ex 417	. . . . . . . . 9 ⊢ (𝜑 → (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
101	100	alrimivv 1958	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
102	88, 101	jca 520	. . . . . . 7 ⊢ (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)))
103		imaeq1 6059	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑠 “ (𝑂 ∪ ran 𝐻)) = (𝑆 “ (𝑂 ∪ ran 𝐻)))
104	103	sseq1d 3969	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
105		fveq1 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑥)))
106	105	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑥))))
107		fveq1 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑦)))
108	107	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑦))))
109	106, 108	xpeq12d 5694	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) = ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))))
110	109	sseq1d 3969	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾 ↔ ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
111	110	imbi2d 343	. . . . . . . . . . . 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 1953	. . . . . . . . . . 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 643	. . . . . . . . . 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 6882	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑃) = (𝑆‘𝑃))
115	114	eleq1d 2848	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ((𝑠‘𝑃) ∈ 𝑐 ↔ (𝑆‘𝑃) ∈ 𝑐))
116	113, 115	imbi12d 347	. . . . . . . . 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 3578	. . . . . . . 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 18	. . . . . . 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 45	. . . . . 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 60	. . . . 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 61	. . . 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 1957	. . 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 6896	. . . 4 ⊢ (𝑆‘𝑃) ∈ V
124	123	elintab 4925	. . 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 237	. 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 2866	1 ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079  Vcvv 3455   ∪ cun 3904   ⊆ wss 3906  ⟨cop 4596  ⟨cotp 4598  ∩ cint 4913   class class class wbr 5110   × cxp 5661  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  Fincfn 8944  mVRcmvar 35931  mAxcmax 35935  mExcmex 35937  mDVcmdv 35938  mVarscmvrs 35939  mSubstcmsub 35941  mVHcmvh 35942  mPreStcmpst 35943  mStatcmsta 35945  mFScmfs 35946  mClscmcls 35947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-word 14553  df-concat 14610  df-s1 14636  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-0g 17495  df-gsum 17496  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-submnd 18843  df-frmd 18909  df-mrex 35956  df-mex 35957  df-mrsub 35960  df-msub 35961  df-mvh 35962  df-mpst 35963  df-msr 35964  df-msta 35965  df-mfs 35966  df-mcls 35967

This theorem is referenced by:  mclsppslem  36053