Theorem mclsppslem 35818

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 35804.) 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 35767	. . . 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 2740	. . . . . . . . 9 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
8		eqid 2740	. . . . . . . . 9 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
9	7, 8	maxsta 35789	. . . . . . . 8 ⊢ (𝑇 ∈ mFS → (mAx‘𝑇) ⊆ (mStat‘𝑇))
10	6, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mStat‘𝑇))
11		eqid 2740	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
12	11, 8	mstapst 35782	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
13	10, 12	sstrdi 3934	. . . . . 6 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mPreSt‘𝑇))
14		mclsppslem.9	. . . . . 6 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
15	13, 14	sseldd 3923	. . . . 5 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
16		mclspps.d	. . . . . 6 ⊢ 𝐷 = (mDV‘𝑇)
17	16, 3, 11	elmpst 35771	. . . . 5 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
18	15, 17	sylib 219	. . . 4 ⊢ (𝜑 → ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
19	18	simp3d 1150	. . 3 ⊢ (𝜑 → 𝑝 ∈ 𝐸)
20	5, 19	ffvelcdmd 7033	. 2 ⊢ (𝜑 → (𝑠‘𝑝) ∈ 𝐸)
21		fvco3 6934	. . . 4 ⊢ ((𝑠:𝐸⟶𝐸 ∧ 𝑝 ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘𝑝) = (𝑆‘(𝑠‘𝑝)))
22	5, 19, 21	syl2anc 590	. . 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 35766	. . . . 5 ⊢ ((𝑆 ∈ ran 𝐿 ∧ 𝑠 ∈ ran 𝐿) → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
31	29, 1, 30	syl2anc 590	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
32	2, 3	msubf 35767	. . . . . . . . 9 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
33	29, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
34		fco 6686	. . . . . . . 8 ⊢ ((𝑆:𝐸⟶𝐸 ∧ 𝑠:𝐸⟶𝐸) → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
35	33, 5, 34	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
36	35	ffnd 6663	. . . . . 6 ⊢ (𝜑 → (𝑆 ∘ 𝑠) Fn 𝐸)
37	36	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → (𝑆 ∘ 𝑠) Fn 𝐸)
38		mclsppslem.11	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
39	5	ffund 6666	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑠)
40	17	simp2bi 1152	. . . . . . . . . . . . . 14 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin))
41	15, 40	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin))
42	41	simpld 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑜 ⊆ 𝐸)
43	26, 3, 27	mvhf 35793	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
44		frn 6669	. . . . . . . . . . . . 13 ⊢ (𝐻:𝑉⟶𝐸 → ran 𝐻 ⊆ 𝐸)
45	6, 43, 44	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
46	42, 45	unssd 4128	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ 𝐸)
47	5	fdmd 6672	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑠 = 𝐸)
48	46, 47	sseqtrrd 3959	. . . . . . . . . 10 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠)
49		funimass3 7002	. . . . . . . . . 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 590	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))))
51	38, 50	mpbid 233	. . . . . . . 8 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))))
52		cnvco 5834	. . . . . . . . . 10 ⊢ ◡(𝑆 ∘ 𝑠) = (◡𝑠 ∘ ◡𝑆)
53	52	imaeq1i 6016	. . . . . . . . 9 ⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵))
54		imaco 6209	. . . . . . . . 9 ⊢ ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))
55	53, 54	eqtri 2763	. . . . . . . 8 ⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))
56	51, 55	sseqtrrdi 3963	. . . . . . 7 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
57	56	unssad 4129	. . . . . 6 ⊢ (𝜑 → 𝑜 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
58	57	sselda 3922	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
59		elpreima 7006	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → (𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ (𝑐 ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))))
60	59	simplbda 500	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
61	37, 58, 60	syl2anc 590	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
62	36	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑆 ∘ 𝑠) Fn 𝐸)
63	56	unssbd 4130	. . . . . . 7 ⊢ (𝜑 → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
64	63	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
65		ffn 6662	. . . . . . . 8 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
66	6, 43, 65	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn 𝑉)
67		fnfvelrn 7028	. . . . . . 7 ⊢ ((𝐻 Fn 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
68	66, 67	sylan 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
69	64, 68	sseldd 3923	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
70		elpreima 7006	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → ((𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑡) ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))))
71	70	simplbda 500	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
72	62, 69, 71	syl2anc 590	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
73	5	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑠:𝐸⟶𝐸)
74	6, 43	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
75	74	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝐻:𝑉⟶𝐸)
76	18	simp1d 1148	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚))
77	76	simpld 495	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑚 ⊆ 𝐷)
78	26, 16	mdvval 35739	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )
79		difss 4073	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 × 𝑉) ∖ I ) ⊆ (𝑉 × 𝑉)
80	78, 79	eqsstri 3968	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ⊆ (𝑉 × 𝑉)
81	77, 80	sstrdi 3934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑚 ⊆ (𝑉 × 𝑉))
82	81	ssbrd 5122	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑐𝑚𝑑 → 𝑐(𝑉 × 𝑉)𝑑))
83	82	imp 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐(𝑉 × 𝑉)𝑑)
84		brxp 5674	. . . . . . . . . . . . . . . 16 ⊢ (𝑐(𝑉 × 𝑉)𝑑 ↔ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
85	83, 84	sylib 219	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
86	85	simpld 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐 ∈ 𝑉)
87	75, 86	ffvelcdmd 7033	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑐) ∈ 𝐸)
88		fvco3 6934	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑐) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
89	73, 87, 88	syl2anc 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
90	89	fveq2d 6838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))))
91	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑇 ∈ mFS)
92	29	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑆 ∈ ran 𝐿)
93	73, 87	ffvelcdmd 7033	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑐)) ∈ 𝐸)
94	2, 3, 28, 27	msubvrs 35795	. . . . . . . . . . . 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 1379	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
96	90, 95	eqtrd 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
97	96	eleq2d 2826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ 𝑎 ∈ ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))))
98		eliun 4932	. . . . . . . . 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 288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))))
100	85	simprd 496	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑑 ∈ 𝑉)
101	75, 100	ffvelcdmd 7033	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑑) ∈ 𝐸)
102		fvco3 6934	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑑) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
103	73, 101, 102	syl2anc 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
104	103	fveq2d 6838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))))
105	73, 101	ffvelcdmd 7033	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑑)) ∈ 𝐸)
106	2, 3, 28, 27	msubvrs 35795	. . . . . . . . . . . 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 1379	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
108	104, 107	eqtrd 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
109	108	eleq2d 2826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ 𝑏 ∈ ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))))
110		eliun 4932	. . . . . . . . 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 288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))
112	99, 111	anbi12d 638	. . . . . . 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 3212	. . . . . . . 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 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝜑)
115		brxp 5674	. . . . . . . . . . . 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 5084	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑧𝑚𝑤 ↔ 𝑐𝑚𝑑))
118		simpl 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑧 = 𝑐)
119	118	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑧) = (𝐻‘𝑐))
120	119	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑧)) = (𝑠‘(𝐻‘𝑐)))
121	120	fveq2d 6838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑧))) = (𝑊‘(𝑠‘(𝐻‘𝑐))))
122		simpr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑤 = 𝑑)
123	122	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑤) = (𝐻‘𝑑))
124	123	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑤)) = (𝑠‘(𝐻‘𝑑)))
125	124	fveq2d 6838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑤))) = (𝑊‘(𝑠‘(𝐻‘𝑑))))
126	121, 125	xpeq12d 5656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) = ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))))
127	126	sseq1d 3953	. . . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . . 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 3545	. . . . . . . . . . . . . . . 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 3439	. . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)
133	132	ssbrd 5122	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑢((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))𝑣 → 𝑢𝑀𝑣))
134	115, 133	biimtrrid 244	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))) → 𝑢𝑀𝑣))
135	134	imp 407	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝑢𝑀𝑣)
136		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
137		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
138		breq12 5084	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥𝑀𝑦 ↔ 𝑢𝑀𝑣))
139		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
140	139	fveq2d 6838	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑥) = (𝐻‘𝑢))
141	140	fveq2d 6838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑢)))
142	141	fveq2d 6838	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑢))))
143	142	eleq2d 2826	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ↔ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))))
144		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
145	144	fveq2d 6838	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑦) = (𝐻‘𝑣))
146	145	fveq2d 6838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑣)))
147	146	fveq2d 6838	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑣))))
148	147	eleq2d 2826	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))) ↔ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))
149	138, 143, 148	3anbi123d 1444	. . . . . . . . . . . . . . 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 636	. . . . . . . . . . . . . 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 342	. . . . . . . . . . . . 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 3513	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) → 𝑎𝐾𝑏)
154	153	3exp2 1361	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢𝑀𝑣 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))) → 𝑎𝐾𝑏))))
155	154	imp4b 422	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢𝑀𝑣) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
156	114, 135, 155	syl2anc 590	. . . . . . . . 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 3197	. . . . . . . 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 244	. . . . . . 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 241	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)))) → 𝑎𝐾𝑏))
160	159	exp4b 431	. . . . 5 ⊢ (𝜑 → (𝑐𝑚𝑑 → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) → 𝑎𝐾𝑏))))
161	160	3imp2 1356	. . . 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 35804	. . 3 ⊢ (𝜑 → ((𝑆 ∘ 𝑠)‘𝑝) ∈ (𝐾𝐶𝐵))
163	22, 162	eqeltrrd 2841	. 2 ⊢ (𝜑 → (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))
164	33	ffnd 6663	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝐸)
165		elpreima 7006	. . 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 719	1 ⊢ (𝜑 → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092  ∀wal 1545   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  Vcvv 3432   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  ⟨cotp 4570  ∪ ciun 4928   class class class wbr 5079   I cid 5519   × cxp 5623  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   “ cima 5628   ∘ ccom 5629  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  Fincfn 8890  mVRcmvar 35696  mAxcmax 35700  mExcmex 35702  mDVcmdv 35703  mVarscmvrs 35704  mSubstcmsub 35706  mVHcmvh 35707  mPreStcmpst 35708  mStatcmsta 35710  mFScmfs 35711  mClscmcls 35712  mPPStcmpps 35713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-pm 8773  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-n0 12436  df-xnn0 12509  df-z 12523  df-uz 12787  df-fz 13460  df-fzo 13607  df-seq 13962  df-hash 14291  df-word 14474  df-lsw 14523  df-concat 14531  df-s1 14557  df-substr 14602  df-pfx 14632  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-0g 17402  df-gsum 17403  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-mhm 18749  df-submnd 18750  df-frmd 18815  df-vrmd 18816  df-mrex 35721  df-mex 35722  df-mdv 35723  df-mvrs 35724  df-mrsub 35725  df-msub 35726  df-mvh 35727  df-mpst 35728  df-msr 35729  df-msta 35730  df-mfs 35731  df-mcls 35732

This theorem is referenced by:  mclspps  35819