mclsppslem - Mathbox for Mario Carneiro

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Theorem mclsppslem 30734

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 30720.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
mclspps.d	⊢ 𝐷 = (mDV‘𝑇)
mclspps.e	⊢ 𝐸 = (mEx‘𝑇)
mclspps.c	⊢ 𝐶 = (mCls‘𝑇)
mclspps.1	⊢ (𝜑 → 𝑇 ∈ mFS)
mclspps.2	⊢ (𝜑 → 𝐾 ⊆ 𝐷)
mclspps.3	⊢ (𝜑 → 𝐵 ⊆ 𝐸)
mclspps.j	⊢ 𝐽 = (mPPSt‘𝑇)
mclspps.l	⊢ 𝐿 = (mSubst‘𝑇)
mclspps.v	⊢ 𝑉 = (mVR‘𝑇)
mclspps.h	⊢ 𝐻 = (mVH‘𝑇)
mclspps.w	⊢ 𝑊 = (mVars‘𝑇)
mclspps.4	⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
mclspps.5	⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
mclspps.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
mclspps.7	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
mclspps.8	⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)
mclsppslem.9	⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
mclsppslem.10	⊢ (𝜑 → 𝑠 ∈ ran 𝐿)
mclsppslem.11	⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (^◡𝑆 “ (𝐾𝐶𝐵)))
mclsppslem.12	⊢ (𝜑 → ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀))

**Assertion**
Ref	Expression
mclsppslem	⊢ (𝜑 → (𝑠‘𝑝) ∈ (^◡𝑆 “ (𝐾𝐶𝐵)))

Distinct variable groups: 𝑚,𝑜,𝑝,𝑠,𝑣,𝐸 𝑎,𝑏,𝑚,𝑜,𝑝,𝑠,𝑣,𝑤,𝑥,𝑦,𝑧,𝐻 𝑣,𝑉,𝑧 𝐾,𝑎,𝑏,𝑚,𝑜,𝑝,𝑠,𝑣,𝑥,𝑦 𝑇,𝑎,𝑏,𝑚,𝑜,𝑝,𝑠,𝑣,𝑤,𝑥,𝑦,𝑧 𝐿,𝑎,𝑏,𝑚,𝑜,𝑝,𝑠,𝑣,𝑤,𝑥,𝑦,𝑧 𝑆,𝑎,𝑏,𝑚,𝑜,𝑝,𝑠,𝑣,𝑥,𝑦 𝐵,𝑎,𝑏,𝑚,𝑜,𝑝,𝑠,𝑣,𝑥,𝑦 𝑊,𝑎,𝑏,𝑚,𝑜,𝑝,𝑠,𝑣,𝑤,𝑥,𝑦,𝑧 𝐶,𝑎,𝑏,𝑚,𝑜,𝑝,𝑠,𝑣,𝑥,𝑦,𝑧 𝑀,𝑎,𝑏,𝑚,𝑜,𝑝,𝑠,𝑣,𝑤,𝑥,𝑦,𝑧 𝑚,𝑂,𝑜,𝑝,𝑠,𝑣,𝑤,𝑥,𝑧 𝜑,𝑎,𝑏,𝑣,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑧,𝑤,𝑚,𝑜,𝑠,𝑝) 𝐵(𝑧,𝑤) 𝐶(𝑤) 𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑚,𝑜,𝑠,𝑝,𝑎,𝑏) 𝑃(𝑥,𝑦,𝑧,𝑤,𝑣,𝑚,𝑜,𝑠,𝑝,𝑎,𝑏) 𝑆(𝑧,𝑤) 𝐸(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏) 𝐽(𝑥,𝑦,𝑧,𝑤,𝑣,𝑚,𝑜,𝑠,𝑝,𝑎,𝑏) 𝐾(𝑧,𝑤) 𝑂(𝑦,𝑎,𝑏) 𝑉(𝑥,𝑦,𝑤,𝑚,𝑜,𝑠,𝑝,𝑎,𝑏)

Proof of Theorem mclsppslem Dummy variables 𝑡 𝑢 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mclsppslem.10	. . . 4 ⊢ (𝜑 → 𝑠 ∈ ran 𝐿)
2		mclspps.l	. . . . 5 ⊢ 𝐿 = (mSubst‘𝑇)
3		mclspps.e	. . . . 5 ⊢ 𝐸 = (mEx‘𝑇)
4	2, 3	msubf 30683	. . . 4 ⊢ (𝑠 ∈ ran 𝐿 → 𝑠:𝐸⟶𝐸)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑠:𝐸⟶𝐸)
6		mclspps.1	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ mFS)
7		eqid 2610	. . . . . . . . 9 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
8		eqid 2610	. . . . . . . . 9 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
9	7, 8	maxsta 30705	. . . . . . . 8 ⊢ (𝑇 ∈ mFS → (mAx‘𝑇) ⊆ (mStat‘𝑇))
10	6, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mStat‘𝑇))
11		eqid 2610	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
12	11, 8	mstapst 30698	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
13	10, 12	syl6ss 3580	. . . . . 6 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mPreSt‘𝑇))
14		mclsppslem.9	. . . . . 6 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
15	13, 14	sseldd 3569	. . . . 5 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
16		mclspps.d	. . . . . 6 ⊢ 𝐷 = (mDV‘𝑇)
17	16, 3, 11	elmpst 30687	. . . . 5 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ^◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
18	15, 17	sylib 207	. . . 4 ⊢ (𝜑 → ((𝑚 ⊆ 𝐷 ∧ ^◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
19	18	simp3d 1068	. . 3 ⊢ (𝜑 → 𝑝 ∈ 𝐸)
20	5, 19	ffvelrnd 6268	. 2 ⊢ (𝜑 → (𝑠‘𝑝) ∈ 𝐸)
21		fvco3 6185	. . . 4 ⊢ ((𝑠:𝐸⟶𝐸 ∧ 𝑝 ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘𝑝) = (𝑆‘(𝑠‘𝑝)))
22	5, 19, 21	syl2anc 691	. . 3 ⊢ (𝜑 → ((𝑆 ∘ 𝑠)‘𝑝) = (𝑆‘(𝑠‘𝑝)))
23		mclspps.c	. . . 4 ⊢ 𝐶 = (mCls‘𝑇)
24		mclspps.2	. . . 4 ⊢ (𝜑 → 𝐾 ⊆ 𝐷)
25		mclspps.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
26		mclspps.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
27		mclspps.h	. . . 4 ⊢ 𝐻 = (mVH‘𝑇)
28		mclspps.w	. . . 4 ⊢ 𝑊 = (mVars‘𝑇)
29		mclspps.5	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
30	2	msubco 30682	. . . . 5 ⊢ ((𝑆 ∈ ran 𝐿 ∧ 𝑠 ∈ ran 𝐿) → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
31	29, 1, 30	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
32	2, 3	msubf 30683	. . . . . . . . 9 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
33	29, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
34		fco 5971	. . . . . . . 8 ⊢ ((𝑆:𝐸⟶𝐸 ∧ 𝑠:𝐸⟶𝐸) → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
35	33, 5, 34	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
36		ffn 5958	. . . . . . 7 ⊢ ((𝑆 ∘ 𝑠):𝐸⟶𝐸 → (𝑆 ∘ 𝑠) Fn 𝐸)
37	35, 36	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑆 ∘ 𝑠) Fn 𝐸)
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → (𝑆 ∘ 𝑠) Fn 𝐸)
39		mclsppslem.11	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (^◡𝑆 “ (𝐾𝐶𝐵)))
40		ffun 5961	. . . . . . . . . . 11 ⊢ (𝑠:𝐸⟶𝐸 → Fun 𝑠)
41	5, 40	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑠)
42	17	simp2bi 1070	. . . . . . . . . . . . . 14 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin))
43	15, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin))
44	43	simpld 474	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑜 ⊆ 𝐸)
45	26, 3, 27	mvhf 30709	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
46		frn 5966	. . . . . . . . . . . . 13 ⊢ (𝐻:𝑉⟶𝐸 → ran 𝐻 ⊆ 𝐸)
47	6, 45, 46	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
48	44, 47	unssd 3751	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ 𝐸)
49		fdm 5964	. . . . . . . . . . . 12 ⊢ (𝑠:𝐸⟶𝐸 → dom 𝑠 = 𝐸)
50	5, 49	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑠 = 𝐸)
51	48, 50	sseqtr4d 3605	. . . . . . . . . 10 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠)
52		funimass3 6241	. . . . . . . . . 10 ⊢ ((Fun 𝑠 ∧ (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (^◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑜 ∪ ran 𝐻) ⊆ (^◡𝑠 “ (^◡𝑆 “ (𝐾𝐶𝐵)))))
53	41, 51, 52	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (^◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑜 ∪ ran 𝐻) ⊆ (^◡𝑠 “ (^◡𝑆 “ (𝐾𝐶𝐵)))))
54	39, 53	mpbid 221	. . . . . . . 8 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (^◡𝑠 “ (^◡𝑆 “ (𝐾𝐶𝐵))))
55		cnvco 5230	. . . . . . . . . 10 ⊢ ^◡(𝑆 ∘ 𝑠) = (^◡𝑠 ∘ ^◡𝑆)
56	55	imaeq1i 5382	. . . . . . . . 9 ⊢ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = ((^◡𝑠 ∘ ^◡𝑆) “ (𝐾𝐶𝐵))
57		imaco 5557	. . . . . . . . 9 ⊢ ((^◡𝑠 ∘ ^◡𝑆) “ (𝐾𝐶𝐵)) = (^◡𝑠 “ (^◡𝑆 “ (𝐾𝐶𝐵)))
58	56, 57	eqtri 2632	. . . . . . . 8 ⊢ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = (^◡𝑠 “ (^◡𝑆 “ (𝐾𝐶𝐵)))
59	54, 58	syl6sseqr 3615	. . . . . . 7 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
60	59	unssad 3752	. . . . . 6 ⊢ (𝜑 → 𝑜 ⊆ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
61	60	sselda 3568	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → 𝑐 ∈ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
62		elpreima 6245	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → (𝑐 ∈ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ (𝑐 ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))))
63	62	simplbda 652	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ 𝑐 ∈ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
64	38, 61, 63	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
65	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑆 ∘ 𝑠) Fn 𝐸)
66	59	unssbd 3753	. . . . . . 7 ⊢ (𝜑 → ran 𝐻 ⊆ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
67	66	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ran 𝐻 ⊆ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
68		ffn 5958	. . . . . . . 8 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
69	6, 45, 68	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn 𝑉)
70		fnfvelrn 6264	. . . . . . 7 ⊢ ((𝐻 Fn 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
71	69, 70	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
72	67, 71	sseldd 3569	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
73		elpreima 6245	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → ((𝐻‘𝑡) ∈ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑡) ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))))
74	73	simplbda 652	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ (𝐻‘𝑡) ∈ (^◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
75	65, 72, 74	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
76	5	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑠:𝐸⟶𝐸)
77	6, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
78	77	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝐻:𝑉⟶𝐸)
79	18	simp1d 1066	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑚 ⊆ 𝐷 ∧ ^◡𝑚 = 𝑚))
80	79	simpld 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑚 ⊆ 𝐷)
81	26, 16	mdvval 30655	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )
82		difss 3699	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 × 𝑉) ∖ I ) ⊆ (𝑉 × 𝑉)
83	81, 82	eqsstri 3598	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ⊆ (𝑉 × 𝑉)
84	80, 83	syl6ss 3580	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑚 ⊆ (𝑉 × 𝑉))
85	84	ssbrd 4626	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑐𝑚𝑑 → 𝑐(𝑉 × 𝑉)𝑑))
86	85	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐(𝑉 × 𝑉)𝑑)
87		brxp 5071	. . . . . . . . . . . . . . . 16 ⊢ (𝑐(𝑉 × 𝑉)𝑑 ↔ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
88	86, 87	sylib 207	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
89	88	simpld 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐 ∈ 𝑉)
90	78, 89	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑐) ∈ 𝐸)
91		fvco3 6185	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑐) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
92	76, 90, 91	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
93	92	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))))
94	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑇 ∈ mFS)
95	29	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑆 ∈ ran 𝐿)
96	76, 90	ffvelrnd 6268	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑐)) ∈ 𝐸)
97	2, 3, 28, 27	msubvrs 30711	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑆 ∈ ran 𝐿 ∧ (𝑠‘(𝐻‘𝑐)) ∈ 𝐸) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
98	94, 95, 96, 97	syl3anc 1318	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
99	93, 98	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
100	99	eleq2d 2673	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ 𝑎 ∈ ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))))
101		eliun 4460	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))))
102	100, 101	syl6bb 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))))
103	88	simprd 478	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑑 ∈ 𝑉)
104	78, 103	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑑) ∈ 𝐸)
105		fvco3 6185	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑑) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
106	76, 104, 105	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
107	106	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))))
108	76, 104	ffvelrnd 6268	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑑)) ∈ 𝐸)
109	2, 3, 28, 27	msubvrs 30711	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑆 ∈ ran 𝐿 ∧ (𝑠‘(𝐻‘𝑑)) ∈ 𝐸) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
110	94, 95, 108, 109	syl3anc 1318	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
111	107, 110	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
112	111	eleq2d 2673	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ 𝑏 ∈ ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))))
113		eliun 4460	. . . . . . . . 9 ⊢ (𝑏 ∈ ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))
114	112, 113	syl6bb 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))
115	102, 114	anbi12d 743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)))) ↔ (∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))))
116		reeanv 3086	. . . . . . . 8 ⊢ (∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) ↔ (∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))
117		simpll 786	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝜑)
118		brxp 5071	. . . . . . . . . . . 12 ⊢ (𝑢((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))𝑣 ↔ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))))
119		mclsppslem.12	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀))
120		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
121		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑑 ∈ V
122		breq12 4588	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑧𝑚𝑤 ↔ 𝑐𝑚𝑑))
123		simpl 472	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑧 = 𝑐)
124	123	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑧) = (𝐻‘𝑐))
125	124	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑧)) = (𝑠‘(𝐻‘𝑐)))
126	125	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑧))) = (𝑊‘(𝑠‘(𝐻‘𝑐))))
127		simpr 476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑤 = 𝑑)
128	127	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑤) = (𝐻‘𝑑))
129	128	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑤)) = (𝑠‘(𝐻‘𝑑)))
130	129	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑤))) = (𝑊‘(𝑠‘(𝐻‘𝑑))))
131	126, 130	xpeq12d 5064	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) = ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))))
132	131	sseq1d 3595	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀 ↔ ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀))
133	122, 132	imbi12d 333	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀) ↔ (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)))
134	133	spc2gv 3269	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ V ∧ 𝑑 ∈ V) → (∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀) → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)))
135	120, 121, 134	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀) → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀))
136	119, 135	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀))
137	136	imp 444	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)
138	137	ssbrd 4626	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑢((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))𝑣 → 𝑢𝑀𝑣))
139	118, 138	syl5bir 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))) → 𝑢𝑀𝑣))
140	139	imp 444	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝑢𝑀𝑣)
141		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
142		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
143		breq12 4588	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥𝑀𝑦 ↔ 𝑢𝑀𝑣))
144		simpl 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
145	144	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑥) = (𝐻‘𝑢))
146	145	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑢)))
147	146	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑢))))
148	147	eleq2d 2673	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ↔ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))))
149		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
150	149	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑦) = (𝐻‘𝑣))
151	150	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑣)))
152	151	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑣))))
153	152	eleq2d 2673	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))) ↔ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))
154	143, 148, 153	3anbi123d 1391	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))) ↔ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))))
155	154	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) ↔ (𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))))
156	155	imbi1d 330	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏) ↔ ((𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) → 𝑎𝐾𝑏)))
157		mclspps.8	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)
158	141, 142, 156, 157	vtocl2 3234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) → 𝑎𝐾𝑏)
159	158	3exp2 1277	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢𝑀𝑣 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))) → 𝑎𝐾𝑏))))
160	159	imp4b 611	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢𝑀𝑣) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
161	117, 140, 160	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
162	161	rexlimdvva 3020	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
163	116, 162	syl5bir 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
164	115, 163	sylbid 229	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)))) → 𝑎𝐾𝑏))
165	164	exp4b 630	. . . . 5 ⊢ (𝜑 → (𝑐𝑚𝑑 → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) → 𝑎𝐾𝑏))))
166	165	3imp2 1274	. . . 4 ⊢ ((𝜑 ∧ (𝑐𝑚𝑑 ∧ 𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))))) → 𝑎𝐾𝑏)
167	16, 3, 23, 6, 24, 25, 7, 2, 26, 27, 28, 14, 31, 64, 75, 166	mclsax 30720	. . 3 ⊢ (𝜑 → ((𝑆 ∘ 𝑠)‘𝑝) ∈ (𝐾𝐶𝐵))
168	22, 167	eqeltrrd 2689	. 2 ⊢ (𝜑 → (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))
169		ffn 5958	. . . 4 ⊢ (𝑆:𝐸⟶𝐸 → 𝑆 Fn 𝐸)
170	33, 169	syl 17	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝐸)
171		elpreima 6245	. . 3 ⊢ (𝑆 Fn 𝐸 → ((𝑠‘𝑝) ∈ (^◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))))
172	170, 171	syl 17	. 2 ⊢ (𝜑 → ((𝑠‘𝑝) ∈ (^◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))))
173	20, 168, 172	mpbir2and 959	1 ⊢ (𝜑 → (𝑠‘𝑝) ∈ (^◡𝑆 “ (𝐾𝐶𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ⟨cotp 4133 ∪ ciun 4455 class class class wbr 4583 I cid 4948 × cxp 5036 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ∘ ccom 5042 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 mVRcmvar 30612 mAxcmax 30616 mExcmex 30618 mDVcmdv 30619 mVarscmvrs 30620 mSubstcmsub 30622 mVHcmvh 30623 mPreStcmpst 30624 mStatcmsta 30626 mFScmfs 30627 mClscmcls 30628 mPPStcmpps 30629

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-frmd 17209 df-vrmd 17210 df-mrex 30637 df-mex 30638 df-mdv 30639 df-mvrs 30640 df-mrsub 30641 df-msub 30642 df-mvh 30643 df-mpst 30644 df-msr 30645 df-msta 30646 df-mfs 30647 df-mcls 30648

This theorem is referenced by: mclspps 30735

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