Theorem mclsppslem 35588

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 35574.) 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 35537	. . . 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 2737	. . . . . . . . 9 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
8		eqid 2737	. . . . . . . . 9 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
9	7, 8	maxsta 35559	. . . . . . . 8 ⊢ (𝑇 ∈ mFS → (mAx‘𝑇) ⊆ (mStat‘𝑇))
10	6, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mStat‘𝑇))
11		eqid 2737	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
12	11, 8	mstapst 35552	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
13	10, 12	sstrdi 3996	. . . . . 6 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mPreSt‘𝑇))
14		mclsppslem.9	. . . . . 6 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
15	13, 14	sseldd 3984	. . . . 5 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
16		mclspps.d	. . . . . 6 ⊢ 𝐷 = (mDV‘𝑇)
17	16, 3, 11	elmpst 35541	. . . . 5 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
18	15, 17	sylib 218	. . . 4 ⊢ (𝜑 → ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
19	18	simp3d 1145	. . 3 ⊢ (𝜑 → 𝑝 ∈ 𝐸)
20	5, 19	ffvelcdmd 7105	. 2 ⊢ (𝜑 → (𝑠‘𝑝) ∈ 𝐸)
21		fvco3 7008	. . . 4 ⊢ ((𝑠:𝐸⟶𝐸 ∧ 𝑝 ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘𝑝) = (𝑆‘(𝑠‘𝑝)))
22	5, 19, 21	syl2anc 584	. . 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 35536	. . . . 5 ⊢ ((𝑆 ∈ ran 𝐿 ∧ 𝑠 ∈ ran 𝐿) → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
31	29, 1, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
32	2, 3	msubf 35537	. . . . . . . . 9 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
33	29, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
34		fco 6760	. . . . . . . 8 ⊢ ((𝑆:𝐸⟶𝐸 ∧ 𝑠:𝐸⟶𝐸) → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
35	33, 5, 34	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
36	35	ffnd 6737	. . . . . 6 ⊢ (𝜑 → (𝑆 ∘ 𝑠) Fn 𝐸)
37	36	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → (𝑆 ∘ 𝑠) Fn 𝐸)
38		mclsppslem.11	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
39	5	ffund 6740	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑠)
40	17	simp2bi 1147	. . . . . . . . . . . . . 14 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin))
41	15, 40	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin))
42	41	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑜 ⊆ 𝐸)
43	26, 3, 27	mvhf 35563	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
44		frn 6743	. . . . . . . . . . . . 13 ⊢ (𝐻:𝑉⟶𝐸 → ran 𝐻 ⊆ 𝐸)
45	6, 43, 44	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
46	42, 45	unssd 4192	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ 𝐸)
47	5	fdmd 6746	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑠 = 𝐸)
48	46, 47	sseqtrrd 4021	. . . . . . . . . 10 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠)
49		funimass3 7074	. . . . . . . . . 10 ⊢ ((Fun 𝑠 ∧ (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))))
50	39, 48, 49	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))))
51	38, 50	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))))
52		cnvco 5896	. . . . . . . . . 10 ⊢ ◡(𝑆 ∘ 𝑠) = (◡𝑠 ∘ ◡𝑆)
53	52	imaeq1i 6075	. . . . . . . . 9 ⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵))
54		imaco 6271	. . . . . . . . 9 ⊢ ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))
55	53, 54	eqtri 2765	. . . . . . . 8 ⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))
56	51, 55	sseqtrrdi 4025	. . . . . . 7 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
57	56	unssad 4193	. . . . . 6 ⊢ (𝜑 → 𝑜 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
58	57	sselda 3983	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
59		elpreima 7078	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → (𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ (𝑐 ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))))
60	59	simplbda 499	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
61	37, 58, 60	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
62	36	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑆 ∘ 𝑠) Fn 𝐸)
63	56	unssbd 4194	. . . . . . 7 ⊢ (𝜑 → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
64	63	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
65		ffn 6736	. . . . . . . 8 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
66	6, 43, 65	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn 𝑉)
67		fnfvelrn 7100	. . . . . . 7 ⊢ ((𝐻 Fn 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
68	66, 67	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
69	64, 68	sseldd 3984	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
70		elpreima 7078	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → ((𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑡) ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))))
71	70	simplbda 499	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
72	62, 69, 71	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
73	5	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑠:𝐸⟶𝐸)
74	6, 43	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
75	74	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝐻:𝑉⟶𝐸)
76	18	simp1d 1143	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚))
77	76	simpld 494	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑚 ⊆ 𝐷)
78	26, 16	mdvval 35509	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )
79		difss 4136	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 × 𝑉) ∖ I ) ⊆ (𝑉 × 𝑉)
80	78, 79	eqsstri 4030	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ⊆ (𝑉 × 𝑉)
81	77, 80	sstrdi 3996	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑚 ⊆ (𝑉 × 𝑉))
82	81	ssbrd 5186	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑐𝑚𝑑 → 𝑐(𝑉 × 𝑉)𝑑))
83	82	imp 406	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐(𝑉 × 𝑉)𝑑)
84		brxp 5734	. . . . . . . . . . . . . . . 16 ⊢ (𝑐(𝑉 × 𝑉)𝑑 ↔ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
85	83, 84	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
86	85	simpld 494	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐 ∈ 𝑉)
87	75, 86	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑐) ∈ 𝐸)
88		fvco3 7008	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑐) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
89	73, 87, 88	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
90	89	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))))
91	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑇 ∈ mFS)
92	29	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑆 ∈ ran 𝐿)
93	73, 87	ffvelcdmd 7105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑐)) ∈ 𝐸)
94	2, 3, 28, 27	msubvrs 35565	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑆 ∈ ran 𝐿 ∧ (𝑠‘(𝐻‘𝑐)) ∈ 𝐸) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
95	91, 92, 93, 94	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
96	90, 95	eqtrd 2777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
97	96	eleq2d 2827	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ 𝑎 ∈ ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))))
98		eliun 4995	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))))
99	97, 98	bitrdi 287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))))
100	85	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑑 ∈ 𝑉)
101	75, 100	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑑) ∈ 𝐸)
102		fvco3 7008	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑑) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
103	73, 101, 102	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
104	103	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))))
105	73, 101	ffvelcdmd 7105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑑)) ∈ 𝐸)
106	2, 3, 28, 27	msubvrs 35565	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ 𝑆 ∈ ran 𝐿 ∧ (𝑠‘(𝐻‘𝑑)) ∈ 𝐸) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
107	91, 92, 105, 106	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
108	104, 107	eqtrd 2777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
109	108	eleq2d 2827	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ 𝑏 ∈ ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))))
110		eliun 4995	. . . . . . . . 9 ⊢ (𝑏 ∈ ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))
111	109, 110	bitrdi 287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))
112	99, 111	anbi12d 632	. . . . . . 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))))))
113		reeanv 3229	. . . . . . . 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)))))
114		simpll 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝜑)
115		brxp 5734	. . . . . . . . . . . 12 ⊢ (𝑢((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))𝑣 ↔ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))))
116		mclsppslem.12	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀))
117		breq12 5148	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑧𝑚𝑤 ↔ 𝑐𝑚𝑑))
118		simpl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑧 = 𝑐)
119	118	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑧) = (𝐻‘𝑐))
120	119	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑧)) = (𝑠‘(𝐻‘𝑐)))
121	120	fveq2d 6910	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑧))) = (𝑊‘(𝑠‘(𝐻‘𝑐))))
122		simpr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑤 = 𝑑)
123	122	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑤) = (𝐻‘𝑑))
124	123	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑤)) = (𝑠‘(𝐻‘𝑑)))
125	124	fveq2d 6910	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑤))) = (𝑊‘(𝑠‘(𝐻‘𝑑))))
126	121, 125	xpeq12d 5716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) = ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))))
127	126	sseq1d 4015	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀 ↔ ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀))
128	117, 127	imbi12d 344	. . . . . . . . . . . . . . . . 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)))
129	128	spc2gv 3600	. . . . . . . . . . . . . . . 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)))
130	129	el2v 3487	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀) → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀))
131	116, 130	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀))
132	131	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)
133	132	ssbrd 5186	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑢((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))𝑣 → 𝑢𝑀𝑣))
134	115, 133	biimtrrid 243	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))) → 𝑢𝑀𝑣))
135	134	imp 406	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝑢𝑀𝑣)
136		vex 3484	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
137		vex 3484	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
138		breq12 5148	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥𝑀𝑦 ↔ 𝑢𝑀𝑣))
139		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
140	139	fveq2d 6910	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑥) = (𝐻‘𝑢))
141	140	fveq2d 6910	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑢)))
142	141	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑢))))
143	142	eleq2d 2827	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ↔ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))))
144		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
145	144	fveq2d 6910	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑦) = (𝐻‘𝑣))
146	145	fveq2d 6910	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑣)))
147	146	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑣))))
148	147	eleq2d 2827	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))) ↔ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))
149	138, 143, 148	3anbi123d 1438	. . . . . . . . . . . . . . 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))))))
150	149	anbi2d 630	. . . . . . . . . . . . . 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)))))))
151	150	imbi1d 341	. . . . . . . . . . . . 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)))
152		mclspps.8	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)
153	136, 137, 151, 152	vtocl2 3566	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) → 𝑎𝐾𝑏)
154	153	3exp2 1355	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢𝑀𝑣 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))) → 𝑎𝐾𝑏))))
155	154	imp4b 421	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢𝑀𝑣) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
156	114, 135, 155	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
157	156	rexlimdvva 3213	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
158	113, 157	biimtrrid 243	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
159	112, 158	sylbid 240	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)))) → 𝑎𝐾𝑏))
160	159	exp4b 430	. . . . 5 ⊢ (𝜑 → (𝑐𝑚𝑑 → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) → 𝑎𝐾𝑏))))
161	160	3imp2 1350	. . . 4 ⊢ ((𝜑 ∧ (𝑐𝑚𝑑 ∧ 𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))))) → 𝑎𝐾𝑏)
162	16, 3, 23, 6, 24, 25, 7, 2, 26, 27, 28, 14, 31, 61, 72, 161	mclsax 35574	. . 3 ⊢ (𝜑 → ((𝑆 ∘ 𝑠)‘𝑝) ∈ (𝐾𝐶𝐵))
163	22, 162	eqeltrrd 2842	. 2 ⊢ (𝜑 → (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))
164	33	ffnd 6737	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝐸)
165		elpreima 7078	. . 3 ⊢ (𝑆 Fn 𝐸 → ((𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))))
166	164, 165	syl 17	. 2 ⊢ (𝜑 → ((𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))))
167	20, 163, 166	mpbir2and 713	1 ⊢ (𝜑 → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ⊆ wss 3951  ⟨cotp 4634  ∪ ciun 4991   class class class wbr 5143   I cid 5577   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688   ∘ ccom 5689  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Fincfn 8985  mVRcmvar 35466  mAxcmax 35470  mExcmex 35472  mDVcmdv 35473  mVarscmvrs 35474  mSubstcmsub 35476  mVHcmvh 35477  mPreStcmpst 35478  mStatcmsta 35480  mFScmfs 35481  mClscmcls 35482  mPPStcmpps 35483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-frmd 18862  df-vrmd 18863  df-mrex 35491  df-mex 35492  df-mdv 35493  df-mvrs 35494  df-mrsub 35495  df-msub 35496  df-mvh 35497  df-mpst 35498  df-msr 35499  df-msta 35500  df-mfs 35501  df-mcls 35502

This theorem is referenced by:  mclspps  35589