Theorem mclsax 35601

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 2713	. . . . . . . 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 4913	. . . . . . . 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 35595	. . . . . . . 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 3966	. . . . . . 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 3941	. . . . . . . . . . . . . . 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 3146	. . . . . . . . . . 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 2731	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
29		eqid 2731	. . . . . . . 8 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
30	28, 29	mstapst 35579	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
31	11, 29	maxsta 35586	. . . . . . . . 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 3935	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mStat‘𝑇))
35	30, 34	sselid 3932	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
36	28	mpstrcl 35573	. . . . . 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 4840	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ⟨𝑚, 𝑜, 𝑝⟩ = ⟨𝑀, 𝑂, 𝑃⟩)
41	40	eleq1d 2816	. . . . . . . 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 6009	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑠 “ (𝑜 ∪ ran 𝐻)) = (𝑠 “ (𝑂 ∪ ran 𝐻)))
44	43	sseq1d 3966	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
45	37	breqd 5102	. . . . . . . . . . . . 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 6826	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑜 = 𝑂 ∧ 𝑝 = 𝑃) → (𝑠‘𝑝) = (𝑠‘𝑃))
50	49	eleq1d 2816	. . . . . . . . . 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 3155	. . . . . . . 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 3559	. . . . . 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 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 3124	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
60		mclsax.v	. . . . . . . . . . . . . . . . 17 ⊢ 𝑉 = (mVR‘𝑇)
61	60, 5, 10	mvhf 35590	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
62	7, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
63		ffn 6651	. . . . . . . . . . . . . . 15 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
64		fveq2 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝐻‘𝑣) → (𝑆‘𝑥) = (𝑆‘(𝐻‘𝑣)))
65	64	eleq1d 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐻‘𝑣) → ((𝑆‘𝑥) ∈ (𝐾𝐶𝐵) ↔ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)))
66	65	ralrn 7021	. . . . . . . . . . . . . . 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 3224	. . . . . . . . . . . 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 3124	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (𝑂 ∪ ran 𝐻)(𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
73		mclsax.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
74	12, 5	msubf 35564	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
75	73, 74	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
76	75	ffund 6655	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑆)
77	4, 5, 28	elmpst 35568	. . . . . . . . . . . . . . 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 6661	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑆 = 𝐸)
82	80, 81	sseqtrrd 3972	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
83	62	frnd 6659	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
84	83, 81	sseqtrrd 3972	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐻 ⊆ dom 𝑆)
85	82, 84	unssd 4142	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂 ∪ ran 𝐻) ⊆ dom 𝑆)
86		funimass4 6886	. . . . . . . . . 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 3173	. . . . . . . . . . 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 2819	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾))
95		df-br 5092	. . . . . . . . . . . . . 14 ⊢ (𝑎𝐾𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝐾)
96	94, 95	bitr4di 289	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → (𝑧 ∈ 𝐾 ↔ 𝑎𝐾𝑏))
97	96	ralxp 5781	. . . . . . . . . . . 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 6004	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑠 “ (𝑂 ∪ ran 𝐻)) = (𝑆 “ (𝑂 ∪ ran 𝐻)))
104	103	sseq1d 3966	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → ((𝑠 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵) ↔ (𝑆 “ (𝑂 ∪ ran 𝐻)) ⊆ (𝐾𝐶𝐵)))
105		fveq1 6821	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑥)))
106	105	fveq2d 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑥))))
107		fveq1 6821	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑦)))
108	107	fveq2d 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑆 → (𝑊‘(𝑠‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑦))))
109	106, 108	xpeq12d 5647	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) = ((𝑊‘(𝑆‘(𝐻‘𝑥))) × (𝑊‘(𝑆‘(𝐻‘𝑦)))))
110	109	sseq1d 3966	. . . . . . . . . . . . 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 6821	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑃) = (𝑆‘𝑃))
115	114	eleq1d 2816	. . . . . . . . . 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 3573	. . . . . . . 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 6835	. . . 4 ⊢ (𝑆‘𝑃) ∈ V
124	123	elintab 4909	. . 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 2834	1 ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  Vcvv 3436   ∪ cun 3900   ⊆ wss 3902  ⟨cop 4582  ⟨cotp 4584  ∩ cint 4897   class class class wbr 5091   × cxp 5614  ^◡ccnv 5615  dom cdm 5616  ran crn 5617   “ cima 5619  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Fincfn 8869  mVRcmvar 35493  mAxcmax 35497  mExcmex 35499  mDVcmdv 35500  mVarscmvrs 35501  mSubstcmsub 35503  mVHcmvh 35504  mPreStcmpst 35505  mStatcmsta 35507  mFScmfs 35508  mClscmcls 35509

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-ot 4585  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-n0 12379  df-z 12466  df-uz 12730  df-fz 13405  df-fzo 13552  df-seq 13906  df-hash 14235  df-word 14418  df-concat 14475  df-s1 14501  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-0g 17342  df-gsum 17343  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-submnd 18689  df-frmd 18754  df-mrex 35518  df-mex 35519  df-mrsub 35522  df-msub 35523  df-mvh 35524  df-mpst 35525  df-msr 35526  df-msta 35527  df-mfs 35528  df-mcls 35529

This theorem is referenced by:  mclsppslem  35615