Theorem mclsax 36156

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 2744	. . . . . . . 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 4926	. . . . . . . 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 36150	. . . . . . . 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 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 3941	. . . . . . . . . . . . . . 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 623	. . . . . . . . . . . . 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 3178	. . . . . . . . . . 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 1949	. . . . . . . . 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 1951	. . . . . . . 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 487	. . . . . 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 2762	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
29		eqid 2762	. . . . . . . 8 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
30	28, 29	mstapst 36134	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
31	11, 29	maxsta 36141	. . . . . . . . 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 3935	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
35	30, 34	sselid 3932	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
36	28	mpstrcl 36128	. . . . . 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 4851	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
41	40	eleq1d 2847	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴))
42	38	uneq1d 4117	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑜 ∪ ran 𝐻) = (𝑂 ∪ ran 𝐻))
43	42	imaeq2d 6060	. . . . . . . . . . . 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 5118	. . . . . . . . . . . . 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 1956	. . . . . . . . . . 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 644	. . . . . . . . . 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 6886	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑠‘𝑝) = (𝑠‘𝑃))
50	49	eleq1d 2847	. . . . . . . . . 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 3187	. . . . . . . 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 3561	. . . . . 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 4103	. . . . . . . . . . 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 3156	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
60		mclsax.v	. . . . . . . . . . . . . . . . 17 ⊢ 𝑉 = (mVR‘𝑇)
61	60, 5, 10	mvhf 36145	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
62	7, 61	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
63		ffn 6706	. . . . . . . . . . . . . . 15 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
64		fveq2 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝐻‘𝑣) → (𝑆‘𝑥) = (𝑆‘(𝐻‘𝑣)))
65	64	eleq1d 2847	. . . . . . . . . . . . . . . 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 3256	. . . . . . . . . . . 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 606	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑂 ∪ ran 𝐻)) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
72	71	ralrimiva 3156	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
73		mclsax.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
74	12, 5	msubf 36119	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
75	73, 74	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
76	75	ffund 6711	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑆)
77	4, 5, 28	elmpst 36123	. . . . . . . . . . . . . . 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 500	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ⊆ 𝐸)
81	75	fdmd 6717	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑆 = 𝐸)
82	80, 81	sseqtrrd 3971	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
83	62	frnd 6715	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
84	83, 81	sseqtrrd 3971	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐻 ⊆ dom 𝑆)
85	82, 84	unssd 4141	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆)
86		funimass4 6946	. . . . . . . . . 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 596	. . . . . . . . 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 427	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))) → 𝑎𝐾𝑏))
92	91	ralrimivv 3205	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑀𝑦) → ∀𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥)))∀𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))𝑎𝐾𝑏)
93		dfss3 3923	. . . . . . . . . . . 12 ⊢ (((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾 ↔ ∀𝑧 ∈ ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦))))𝑧 ∈ 𝐾)
94		eleq1 2850	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
95		df-br 5108	. . . . . . . . . . . . . 14 ⊢ (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
96	94, 95	bitr4di 292	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ 𝑎𝐾𝑏))
97	96	ralxp 5825	. . . . . . . . . . . 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 418	. . . . . . . . 9 ⊢ (𝜑 → (𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
101	100	alrimivv 1961	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾))
102	88, 101	jca 521	. . . . . . 7 ⊢ (𝜑 → ((𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ∧ ∀𝑥∀𝑦(𝑥𝑀𝑦 → ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))) ⊆ 𝐾)))
103		imaeq1 6055	. . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑥)))
106	105	fveq2d 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑥))))
107		fveq1 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑦)))
108	107	fveq2d 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑦))))
109	106, 108	xpeq12d 5690	. . . . . . . . . . . . . 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 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 1956	. . . . . . . . . . 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 644	. . . . . . . . . 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 6881	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑃) = (𝑆‘𝑃))
115	114	eleq1d 2847	. . . . . . . . . 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 3575	. . . . . . . 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 1960	. . 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 6895	. . . 4 ⊢ (𝑆‘𝑃) ∈ V
124	123	elintab 4922	. . 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 2865	1 ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  ⟨cop 4593  ⟨cotp 4595  ∩ cint 4910   class class class wbr 5107   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  Fincfn 8956  mVRcmvar 36048  mAxcmax 36052  mExcmex 36054  mDVcmdv 36055  mVarscmvrs 36056  mSubstcmsub 36058  mVHcmvh 36059  mPreStcmpst 36060  mStatcmsta 36062  mFScmfs 36063  mClscmcls 36064

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-pm 8833  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-fzo 13714  df-seq 14070  df-hash 14399  df-word 14583  df-concat 14640  df-s1 14667  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-0g 17532  df-gsum 17533  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-submnd 18898  df-frmd 18964  df-mrex 36073  df-mex 36074  df-mrsub 36077  df-msub 36078  df-mvh 36079  df-mpst 36080  df-msr 36081  df-msta 36082  df-mfs 36083  df-mcls 36084

This theorem is used by:  mclsppslem  36170