Theorem mclsppslem 36096

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 36082.) 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 36045	. . . 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 2766	. . . . . . . . 9 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
8		eqid 2766	. . . . . . . . 9 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
9	7, 8	maxsta 36067	. . . . . . . 8 ⊢ (𝑇 ∈ mFS → (mAx‘𝑇) ⊆ (mStat‘𝑇))
10	6, 9	syl 18	. . . . . . 7 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mStat‘𝑇))
11		eqid 2766	. . . . . . . 8 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
12	11, 8	mstapst 36060	. . . . . . 7 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
13	10, 12	sstrdi 3952	. . . . . 6 ⊢ (𝜑 → (mAx‘𝑇) ⊆ (mPreSt‘𝑇))
14		mclsppslem.9	. . . . . 6 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
15	13, 14	sseldd 3941	. . . . 5 ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
16		mclspps.d	. . . . . 6 ⊢ 𝐷 = (mDV‘𝑇)
17	16, 3, 11	elmpst 36049	. . . . 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 7087	. 2 ⊢ (𝜑 → (𝑠‘𝑝) ∈ 𝐸)
21		fvco3 6988	. . . 4 ⊢ ((𝑠:𝐸⟶𝐸 ∧ 𝑝 ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘𝑝) = (𝑆‘(𝑠‘𝑝)))
22	5, 19, 21	syl2anc 596	. . 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 36044	. . . . 5 ⊢ ((𝑆 ∈ ran 𝐿 ∧ 𝑠 ∈ ran 𝐿) → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
31	29, 1, 30	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑠) ∈ ran 𝐿)
32	2, 3	msubf 36045	. . . . . . . . 9 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
33	29, 32	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
34		fco 6737	. . . . . . . 8 ⊢ ((𝑆:𝐸⟶𝐸 ∧ 𝑠:𝐸⟶𝐸) → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
35	33, 5, 34	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∘ 𝑠):𝐸⟶𝐸)
36	35	ffnd 6713	. . . . . 6 ⊢ (𝜑 → (𝑆 ∘ 𝑠) Fn 𝐸)
37	36	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → (𝑆 ∘ 𝑠) Fn 𝐸)
38		mclsppslem.11	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
39	5	ffund 6717	. . . . . . . . . 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 500	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑜 ⊆ 𝐸)
43	26, 3, 27	mvhf 36071	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
44		frn 6720	. . . . . . . . . . . . 13 ⊢ (𝐻:𝑉⟶𝐸 → ran 𝐻 ⊆ 𝐸)
45	6, 43, 44	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
46	42, 45	unssd 4148	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ 𝐸)
47	5	fdmd 6723	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑠 = 𝐸)
48	46, 47	sseqtrrd 3977	. . . . . . . . . 10 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠)
49		funimass3 7056	. . . . . . . . . 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 596	. . . . . . . . 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 5880	. . . . . . . . . 10 ⊢ ◡(𝑆 ∘ 𝑠) = (◡𝑠 ∘ ◡𝑆)
53	52	imaeq1i 6064	. . . . . . . . 9 ⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵))
54		imaco 6257	. . . . . . . . 9 ⊢ ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))
55	53, 54	eqtri 2789	. . . . . . . 8 ⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))
56	51, 55	sseqtrrdi 3981	. . . . . . 7 ⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
57	56	unssad 4149	. . . . . 6 ⊢ (𝜑 → 𝑜 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
58	57	sselda 3940	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
59		elpreima 7060	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → (𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ (𝑐 ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))))
60	59	simplbda 505	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
61	37, 58, 60	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵))
62	36	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑆 ∘ 𝑠) Fn 𝐸)
63	56	unssbd 4150	. . . . . . 7 ⊢ (𝜑 → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
64	63	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
65		ffn 6712	. . . . . . . 8 ⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉)
66	6, 43, 65	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn 𝑉)
67		fnfvelrn 7082	. . . . . . 7 ⊢ ((𝐻 Fn 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
68	66, 67	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻)
69	64, 68	sseldd 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)))
70		elpreima 7060	. . . . . 6 ⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → ((𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑡) ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))))
71	70	simplbda 505	. . . . 5 ⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
72	62, 69, 71	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵))
73	5	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑠:𝐸⟶𝐸)
74	6, 43	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
75	74	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝐻:𝑉⟶𝐸)
76	18	simp1d 1160	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚))
77	76	simpld 500	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑚 ⊆ 𝐷)
78	26, 16	mdvval 36017	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I )
79		difss 4093	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 × 𝑉) ∖ I ) ⊆ (𝑉 × 𝑉)
80	78, 79	eqsstri 3986	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ⊆ (𝑉 × 𝑉)
81	77, 80	sstrdi 3952	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑚 ⊆ (𝑉 × 𝑉))
82	81	ssbrd 5159	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑐𝑚𝑑 → 𝑐(𝑉 × 𝑉)𝑑))
83	82	imp 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐(𝑉 × 𝑉)𝑑)
84		brxp 5715	. . . . . . . . . . . . . . . 16 ⊢ (𝑐(𝑉 × 𝑉)𝑑 ↔ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
85	83, 84	sylib 221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
86	85	simpld 500	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐 ∈ 𝑉)
87	75, 86	ffvelcdmd 7087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑐) ∈ 𝐸)
88		fvco3 6988	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑐) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
89	73, 87, 88	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐))))
90	89	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))))
91	6	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑇 ∈ mFS)
92	29	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑆 ∈ ran 𝐿)
93	73, 87	ffvelcdmd 7087	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑐)) ∈ 𝐸)
94	2, 3, 28, 27	msubvrs 36073	. . . . . . . . . . . 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 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))
97	96	eleq2d 2852	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ 𝑎 ∈ ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))))
98		eliun 4965	. . . . . . . . 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 501	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑑 ∈ 𝑉)
101	75, 100	ffvelcdmd 7087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑑) ∈ 𝐸)
102		fvco3 6988	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑑) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
103	73, 101, 102	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑))))
104	103	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))))
105	73, 101	ffvelcdmd 7087	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑑)) ∈ 𝐸)
106	2, 3, 28, 27	msubvrs 36073	. . . . . . . . . . . 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 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))
109	108	eleq2d 2852	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ 𝑏 ∈ ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))))
110		eliun 4965	. . . . . . . . 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 644	. . . . . . 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 3240	. . . . . . . 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 779	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝜑)
115		brxp 5715	. . . . . . . . . . . 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 5119	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑧𝑚𝑤 ↔ 𝑐𝑚𝑑))
118		simpl 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑧 = 𝑐)
119	118	fveq2d 6892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑧) = (𝐻‘𝑐))
120	119	fveq2d 6892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑧)) = (𝑠‘(𝐻‘𝑐)))
121	120	fveq2d 6892	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑧))) = (𝑊‘(𝑠‘(𝐻‘𝑐))))
122		simpr 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑤 = 𝑑)
123	122	fveq2d 6892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑤) = (𝐻‘𝑑))
124	123	fveq2d 6892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑤)) = (𝑠‘(𝐻‘𝑑)))
125	124	fveq2d 6892	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑤))) = (𝑊‘(𝑠‘(𝐻‘𝑑))))
126	121, 125	xpeq12d 5697	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) = ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))))
127	126	sseq1d 3971	. . . . . . . . . . . . . . . . . 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 3562	. . . . . . . . . . . . . . . 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 3465	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)
133	132	ssbrd 5159	. . . . . . . . . . . 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 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝑢𝑀𝑣)
136		vex 3462	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
137		vex 3462	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
138		breq12 5119	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥𝑀𝑦 ↔ 𝑢𝑀𝑣))
139		simpl 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
140	139	fveq2d 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑥) = (𝐻‘𝑢))
141	140	fveq2d 6892	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑢)))
142	141	fveq2d 6892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑢))))
143	142	eleq2d 2852	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ↔ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))))
144		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
145	144	fveq2d 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑦) = (𝐻‘𝑣))
146	145	fveq2d 6892	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑣)))
147	146	fveq2d 6892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑣))))
148	147	eleq2d 2852	. . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . 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 3534	. . . . . . . . . . . 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 427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢𝑀𝑣) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏))
156	114, 135, 155	syl2anc 596	. . . . . . . . 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 3225	. . . . . . . 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 436	. . . . 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 36082	. . 3 ⊢ (𝜑 → ((𝑆 ∘ 𝑠)‘𝑝) ∈ (𝐾𝐶𝐵))
163	22, 162	eqeltrrd 2867	. 2 ⊢ (𝜑 → (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵))
164	33	ffnd 6713	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝐸)
165		elpreima 7060	. . 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 726	1 ⊢ (𝜑 → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ∃wrex 3092  Vcvv 3458   ∖ cdif 3905   ∪ cun 3906   ⊆ wss 3908  ⟨cotp 4602  ∪ ciun 4961   class class class wbr 5114   I cid 5560   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   “ cima 5669   ∘ ccom 5670  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  Fincfn 8952  mVRcmvar 35974  mAxcmax 35978  mExcmex 35980  mDVcmdv 35981  mVarscmvrs 35982  mSubstcmsub 35984  mVHcmvh 35985  mPreStcmpst 35986  mStatcmsta 35988  mFScmfs 35989  mClscmcls 35990  mPPStcmpps 35991

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-map 8835  df-pm 8836  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-n0 12523  df-xnn0 12596  df-z 12610  df-uz 12881  df-fz 13554  df-fzo 13702  df-seq 14058  df-hash 14387  df-word 14571  df-lsw 14620  df-concat 14628  df-s1 14655  df-substr 14701  df-pfx 14733  df-struct 17232  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-0g 17519  df-gsum 17520  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-mhm 18872  df-submnd 18873  df-frmd 18939  df-vrmd 18940  df-mrex 35999  df-mex 36000  df-mdv 36001  df-mvrs 36002  df-mrsub 36003  df-msub 36004  df-mvh 36005  df-mpst 36006  df-msr 36007  df-msta 36008  df-mfs 36009  df-mcls 36010

This theorem is used by:  mclspps  36097