Theorem mclsppslem 36053

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 36039.) 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 36002	. . . 4 ⊢ (𝑠 ∈ ran 𝐿 → 𝑠:𝐸⟶𝐸)
5	1, 4	syl 18	. . 3 ⊢ (𝜑 → 𝑠:𝐸⟶𝐸)
6		mclspps.1	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ mFS)
7		eqid 2763	. . . . . . . . 9 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
8		eqid 2763	. . . . . . . . 9 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
9	7, 8	maxsta 36024	. . . . . . . 8 ⊢ (𝑇 ∈ mFS → (mAx‘𝑇) ⊆ (mStat‘𝑇))
10	6, 9	syl 18	. . . . . . 7 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mStat‘𝑇))
11		eqid 2763	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
12	11, 8	mstapst 36017	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
13	10, 12	sstrdi 3950	. . . . . 6 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mPreSt‘𝑇))
14		mclsppslem.9	. . . . . 6 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
15	13, 14	sseldd 3939	. . . . 5 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
16		mclspps.d	. . . . . 6 ⊢ 𝐷 = (mDV‘𝑇)
17	16, 3, 11	elmpst 36006	. . . . 5 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
18	15, 17	sylib 221	. . . 4 ⊢ (𝜑 → ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
19	18	simp3d 1162	. . 3 ⊢ (𝜑 → 𝑝 ∈ 𝐸)
20	5, 19	ffvelcdmd 7082	. 2 ⊢ (𝜑 → (𝑠‘𝑝) ∈ 𝐸)
21		fvco3 6983	. . . 4 ⊢ ((𝑠:𝐸⟶𝐸 ∧ 𝑝 ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘𝑝) = (𝑆‘(𝑠‘𝑝)))
22	5, 19, 21	syl2anc 595	. . 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 36001	. . . . 5 ⊢ ((𝑆 ∈ ran 𝐿 ∧ 𝑠 ∈ ran 𝐿) → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
31	29, 1, 30	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
32	2, 3	msubf 36002	. . . . . . . . 9 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
33	29, 32	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
34		fco 6732	. . . . . . . 8 ⊢ ((𝑆:𝐸⟶𝐸 ∧ 𝑠:𝐸⟶𝐸) → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
35	33, 5, 34	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
36	35	ffnd 6708	. . . . . 6 ⊢ (𝜑 → (𝑆 ∘ 𝑠) Fn 𝐸)
37	36	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → (𝑆 ∘ 𝑠) Fn 𝐸)
38		mclsppslem.11	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
39	5	ffund 6712	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑠)
40	17	simp2bi 1164	. . . . . . . . . . . . . 14 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin))
41	15, 40	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin))
42	41	simpld 499	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑜 ⊆ 𝐸)
43	26, 3, 27	mvhf 36028	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
44		frn 6715	. . . . . . . . . . . . 13 ⊢ (𝐻:𝑉⟶𝐸 → ran 𝐻 ⊆ 𝐸)
45	6, 43, 44	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
46	42, 45	unssd 4146	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ 𝐸)
47	5	fdmd 6718	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑠 = 𝐸)
48	46, 47	sseqtrrd 3975	. . . . . . . . . 10 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠)
49		funimass3 7051	. . . . . . . . . 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 595	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))))
51	38, 50	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))))
52		cnvco 5877	. . . . . . . . . 10 ⊢ ◡(𝑆 ∘ 𝑠) = (◡𝑠 ∘ ◡𝑆)
53	52	imaeq1i 6061	. . . . . . . . 9 ⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵))
54		imaco 6254	. . . . . . . . 9 ⊢ ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))
55	53, 54	eqtri 2786	. . . . . . . 8 ⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))
56	51, 55	sseqtrrdi 3979	. . . . . . 7 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
57	56	unssad 4147	. . . . . 6 ⊢ (𝜑 → 𝑜 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
58	57	sselda 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
59		elpreima 7055	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → (𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ (𝑐 ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))))
60	59	simplbda 504	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
61	37, 58, 60	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
62	36	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑆 ∘ 𝑠) Fn 𝐸)
63	56	unssbd 4148	. . . . . . 7 ⊢ (𝜑 → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
64	63	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
65		ffn 6707	. . . . . . . 8 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
66	6, 43, 65	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn 𝑉)
67		fnfvelrn 7077	. . . . . . 7 ⊢ ((𝐻 Fn 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
68	66, 67	sylan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
69	64, 68	sseldd 3939	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
70		elpreima 7055	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → ((𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑡) ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))))
71	70	simplbda 504	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
72	62, 69, 71	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
73	5	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑠:𝐸⟶𝐸)
74	6, 43	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
75	74	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝐻:𝑉⟶𝐸)
76	18	simp1d 1160	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚))
77	76	simpld 499	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑚 ⊆ 𝐷)
78	26, 16	mdvval 35974	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )
79		difss 4091	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 × 𝑉) ∖ I ) ⊆ (𝑉 × 𝑉)
80	78, 79	eqsstri 3984	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ⊆ (𝑉 × 𝑉)
81	77, 80	sstrdi 3950	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑚 ⊆ (𝑉 × 𝑉))
82	81	ssbrd 5155	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑐𝑚𝑑 → 𝑐(𝑉 × 𝑉)𝑑))
83	82	imp 411	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐(𝑉 × 𝑉)𝑑)
84		brxp 5712	. . . . . . . . . . . . . . . 16 ⊢ (𝑐(𝑉 × 𝑉)𝑑 ↔ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
85	83, 84	sylib 221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
86	85	simpld 499	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐 ∈ 𝑉)
87	75, 86	ffvelcdmd 7082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑐) ∈ 𝐸)
88		fvco3 6983	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑐) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
89	73, 87, 88	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
90	89	fveq2d 6887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))))
91	6	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑇 ∈ mFS)
92	29	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑆 ∈ ran 𝐿)
93	73, 87	ffvelcdmd 7082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑐)) ∈ 𝐸)
94	2, 3, 28, 27	msubvrs 36030	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
96	90, 95	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
97	96	eleq2d 2849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ 𝑎 ∈ ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))))
98		eliun 4961	. . . . . . . . 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 290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))))
100	85	simprd 500	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑑 ∈ 𝑉)
101	75, 100	ffvelcdmd 7082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑑) ∈ 𝐸)
102		fvco3 6983	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑑) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
103	73, 101, 102	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
104	103	fveq2d 6887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))))
105	73, 101	ffvelcdmd 7082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑑)) ∈ 𝐸)
106	2, 3, 28, 27	msubvrs 36030	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
108	104, 107	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
109	108	eleq2d 2849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ 𝑏 ∈ ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))))
110		eliun 4961	. . . . . . . . 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 290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))
112	99, 111	anbi12d 643	. . . . . . 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 3237	. . . . . . . 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 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝜑)
115		brxp 5712	. . . . . . . . . . . 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 5115	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑧𝑚𝑤 ↔ 𝑐𝑚𝑑))
118		simpl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑧 = 𝑐)
119	118	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑧) = (𝐻‘𝑐))
120	119	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑧)) = (𝑠‘(𝐻‘𝑐)))
121	120	fveq2d 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑧))) = (𝑊‘(𝑠‘(𝐻‘𝑐))))
122		simpr 489	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑤 = 𝑑)
123	122	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑤) = (𝐻‘𝑑))
124	123	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑤)) = (𝑠‘(𝐻‘𝑑)))
125	124	fveq2d 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑤))) = (𝑊‘(𝑠‘(𝐻‘𝑑))))
126	121, 125	xpeq12d 5694	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) = ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))))
127	126	sseq1d 3969	. . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . 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 3560	. . . . . . . . . . . . . . . 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 3462	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀))
132	131	imp 411	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)
133	132	ssbrd 5155	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑢((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))𝑣 → 𝑢𝑀𝑣))
134	115, 133	biimtrrid 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))) → 𝑢𝑀𝑣))
135	134	imp 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝑢𝑀𝑣)
136		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
137		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
138		breq12 5115	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥𝑀𝑦 ↔ 𝑢𝑀𝑣))
139		simpl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
140	139	fveq2d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑥) = (𝐻‘𝑢))
141	140	fveq2d 6887	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑢)))
142	141	fveq2d 6887	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑢))))
143	142	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ↔ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))))
144		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
145	144	fveq2d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑦) = (𝐻‘𝑣))
146	145	fveq2d 6887	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑣)))
147	146	fveq2d 6887	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑣))))
148	147	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))) ↔ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))
149	138, 143, 148	3anbi123d 1464	. . . . . . . . . . . . . . 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 641	. . . . . . . . . . . . . 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 344	. . . . . . . . . . . . 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 3532	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) → 𝑎𝐾𝑏)
154	153	3exp2 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢𝑀𝑣 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))) → 𝑎𝐾𝑏))))
155	154	imp4b 426	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢𝑀𝑣) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
156	114, 135, 155	syl2anc 595	. . . . . . . . 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 3222	. . . . . . . 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 246	. . . . . . 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 243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)))) → 𝑎𝐾𝑏))
160	159	exp4b 435	. . . . 5 ⊢ (𝜑 → (𝑐𝑚𝑑 → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) → 𝑎𝐾𝑏))))
161	160	3imp2 1368	. . . 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 36039	. . 3 ⊢ (𝜑 → ((𝑆 ∘ 𝑠)‘𝑝) ∈ (𝐾𝐶𝐵))
163	22, 162	eqeltrrd 2864	. 2 ⊢ (𝜑 → (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))
164	33	ffnd 6708	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝐸)
165		elpreima 7055	. . 3 ⊢ (𝑆 Fn 𝐸 → ((𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))))
166	164, 165	syl 18	. 2 ⊢ (𝜑 → ((𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))))
167	20, 163, 166	mpbir2and 725	1 ⊢ (𝜑 → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ⟨cotp 4598  ∪ ciun 4957   class class class wbr 5110   I cid 5557   × cxp 5661  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666   ∘ ccom 5667  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  mPPStcmpps 35948

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-xnn0 12579  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-word 14553  df-lsw 14602  df-concat 14610  df-s1 14636  df-substr 14681  df-pfx 14711  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-mhm 18842  df-submnd 18843  df-frmd 18909  df-vrmd 18910  df-mrex 35956  df-mex 35957  df-mdv 35958  df-mvrs 35959  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:  mclspps  36054