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Theorem xkococnlem 22267

Description: Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

**Hypotheses**
Ref	Expression
xkococn.1	⊢ 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓 ∘ 𝑔))
xkococn.s	⊢ (𝜑 → 𝑆 ∈ 𝑛-Locally Comp)
xkococn.k	⊢ (𝜑 → 𝐾 ⊆ ∪ 𝑅)
xkococn.c	⊢ (𝜑 → (𝑅 ↾_t 𝐾) ∈ Comp)
xkococn.v	⊢ (𝜑 → 𝑉 ∈ 𝑇)
xkococn.a	⊢ (𝜑 → 𝐴 ∈ (𝑆 Cn 𝑇))
xkococn.b	⊢ (𝜑 → 𝐵 ∈ (𝑅 Cn 𝑆))
xkococn.i	⊢ (𝜑 → ((𝐴 ∘ 𝐵) “ 𝐾) ⊆ 𝑉)

**Assertion**
Ref	Expression
xkococnlem	⊢ (𝜑 → ∃𝑧 ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))

Distinct variable groups: 𝑧,𝐴 𝑧,𝐵 𝑓,𝑔,ℎ,𝑧,𝑅 𝑆,𝑓,𝑔,𝑧 ℎ,𝐾,𝑧 𝑇,𝑓,𝑔,ℎ,𝑧 𝑧,𝐹 ℎ,𝑉,𝑧 Allowed substitution hints: 𝜑(𝑧,𝑓,𝑔,ℎ) 𝐴(𝑓,𝑔,ℎ) 𝐵(𝑓,𝑔,ℎ) 𝑆(ℎ) 𝐹(𝑓,𝑔,ℎ) 𝐾(𝑓,𝑔) 𝑉(𝑓,𝑔)

Proof of Theorem xkococnlem Dummy variables 𝑘 𝑎 𝑠 𝑢 𝑣 𝑤 𝑥 𝑦 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xkococn.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑅 Cn 𝑆))
2		xkococn.c	. . . 4 ⊢ (𝜑 → (𝑅 ↾_t 𝐾) ∈ Comp)
3		imacmp 22005	. . . 4 ⊢ ((𝐵 ∈ (𝑅 Cn 𝑆) ∧ (𝑅 ↾_t 𝐾) ∈ Comp) → (𝑆 ↾_t (𝐵 “ 𝐾)) ∈ Comp)
4	1, 2, 3	syl2anc 586	. . 3 ⊢ (𝜑 → (𝑆 ↾_t (𝐵 “ 𝐾)) ∈ Comp)
5		xkococn.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑛-Locally Comp)
6	5	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → 𝑆 ∈ 𝑛-Locally Comp)
7		xkococn.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝑆 Cn 𝑇))
8		xkococn.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑇)
9		cnima 21873	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑆 Cn 𝑇) ∧ 𝑉 ∈ 𝑇) → (^◡𝐴 “ 𝑉) ∈ 𝑆)
10	7, 8, 9	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐴 “ 𝑉) ∈ 𝑆)
11	10	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → (^◡𝐴 “ 𝑉) ∈ 𝑆)
12		imaco 6104	. . . . . . . . . . 11 ⊢ ((𝐴 ∘ 𝐵) “ 𝐾) = (𝐴 “ (𝐵 “ 𝐾))
13		xkococn.i	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 ∘ 𝐵) “ 𝐾) ⊆ 𝑉)
14	12, 13	eqsstrrid 4016	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 “ (𝐵 “ 𝐾)) ⊆ 𝑉)
15		eqid 2821	. . . . . . . . . . . . 13 ⊢ ∪ 𝑆 = ∪ 𝑆
16		eqid 2821	. . . . . . . . . . . . 13 ⊢ ∪ 𝑇 = ∪ 𝑇
17	15, 16	cnf 21854	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑆 Cn 𝑇) → 𝐴:∪ 𝑆⟶∪ 𝑇)
18		ffun 6517	. . . . . . . . . . . 12 ⊢ (𝐴:∪ 𝑆⟶∪ 𝑇 → Fun 𝐴)
19	7, 17, 18	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐴)
20		imassrn 5940	. . . . . . . . . . . . 13 ⊢ (𝐵 “ 𝐾) ⊆ ran 𝐵
21		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑅 = ∪ 𝑅
22	21, 15	cnf 21854	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (𝑅 Cn 𝑆) → 𝐵:∪ 𝑅⟶∪ 𝑆)
23		frn 6520	. . . . . . . . . . . . . 14 ⊢ (𝐵:∪ 𝑅⟶∪ 𝑆 → ran 𝐵 ⊆ ∪ 𝑆)
24	1, 22, 23	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐵 ⊆ ∪ 𝑆)
25	20, 24	sstrid 3978	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 “ 𝐾) ⊆ ∪ 𝑆)
26		fdm 6522	. . . . . . . . . . . . 13 ⊢ (𝐴:∪ 𝑆⟶∪ 𝑇 → dom 𝐴 = ∪ 𝑆)
27	7, 17, 26	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐴 = ∪ 𝑆)
28	25, 27	sseqtrrd 4008	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 “ 𝐾) ⊆ dom 𝐴)
29		funimass3 6824	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ (𝐵 “ 𝐾) ⊆ dom 𝐴) → ((𝐴 “ (𝐵 “ 𝐾)) ⊆ 𝑉 ↔ (𝐵 “ 𝐾) ⊆ (^◡𝐴 “ 𝑉)))
30	19, 28, 29	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 “ (𝐵 “ 𝐾)) ⊆ 𝑉 ↔ (𝐵 “ 𝐾) ⊆ (^◡𝐴 “ 𝑉)))
31	14, 30	mpbid 234	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 “ 𝐾) ⊆ (^◡𝐴 “ 𝑉))
32	31	sselda 3967	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → 𝑥 ∈ (^◡𝐴 “ 𝑉))
33		nlly2i 22084	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑛-Locally Comp ∧ (^◡𝐴 “ 𝑉) ∈ 𝑆 ∧ 𝑥 ∈ (^◡𝐴 “ 𝑉)) → ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))
34	6, 11, 32, 33	syl3anc 1367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))
35		nllytop 22081	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
36	5, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Top)
37	36	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → 𝑆 ∈ Top)
38		imaexg 7620	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑅 Cn 𝑆) → (𝐵 “ 𝐾) ∈ V)
39	1, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 “ 𝐾) ∈ V)
40	39	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → (𝐵 “ 𝐾) ∈ V)
41		simprl 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → 𝑢 ∈ 𝑆)
42		elrestr 16702	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Top ∧ (𝐵 “ 𝐾) ∈ V ∧ 𝑢 ∈ 𝑆) → (𝑢 ∩ (𝐵 “ 𝐾)) ∈ (𝑆 ↾_t (𝐵 “ 𝐾)))
43	37, 40, 41, 42	syl3anc 1367	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵 “ 𝐾)) ∈ (𝑆 ↾_t (𝐵 “ 𝐾)))
44		simprr1 1217	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → 𝑥 ∈ 𝑢)
45		simpllr 774	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝐵 “ 𝐾))
46	44, 45	elind 4171	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝑢 ∩ (𝐵 “ 𝐾)))
47		inss1 4205	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ (𝐵 “ 𝐾)) ⊆ 𝑢
48		elpwi 4548	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉) → 𝑠 ⊆ (^◡𝐴 “ 𝑉))
49	48	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → 𝑠 ⊆ (^◡𝐴 “ 𝑉))
50		elssuni 4868	. . . . . . . . . . . . . . . 16 ⊢ ((^◡𝐴 “ 𝑉) ∈ 𝑆 → (^◡𝐴 “ 𝑉) ⊆ ∪ 𝑆)
51	10, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝐴 “ 𝑉) ⊆ ∪ 𝑆)
52	51	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → (^◡𝐴 “ 𝑉) ⊆ ∪ 𝑆)
53	49, 52	sstrd 3977	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → 𝑠 ⊆ ∪ 𝑆)
54		simprr2 1218	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → 𝑢 ⊆ 𝑠)
55	15	ssntr 21666	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Top ∧ 𝑠 ⊆ ∪ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑢 ⊆ 𝑠)) → 𝑢 ⊆ ((int‘𝑆)‘𝑠))
56	37, 53, 41, 54, 55	syl22anc 836	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → 𝑢 ⊆ ((int‘𝑆)‘𝑠))
57	47, 56	sstrid 3978	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠))
58		simprr3 1219	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → (𝑆 ↾_t 𝑠) ∈ Comp)
59	57, 58	jca 514	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → ((𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp))
60		eleq2 2901	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑢 ∩ (𝐵 “ 𝐾)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑢 ∩ (𝐵 “ 𝐾))))
61		cleq1lem 14342	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑢 ∩ (𝐵 “ 𝐾)) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp) ↔ ((𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
62	60, 61	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑢 ∩ (𝐵 “ 𝐾)) → ((𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)) ↔ (𝑥 ∈ (𝑢 ∩ (𝐵 “ 𝐾)) ∧ ((𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp))))
63	62	rspcev 3623	. . . . . . . . . 10 ⊢ (((𝑢 ∩ (𝐵 “ 𝐾)) ∈ (𝑆 ↾_t (𝐵 “ 𝐾)) ∧ (𝑥 ∈ (𝑢 ∩ (𝐵 “ 𝐾)) ∧ ((𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → ∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
64	43, 46, 59, 63	syl12anc 834	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → ∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
65	64	rexlimdvaa 3285	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)) → (∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp) → ∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp))))
66	65	reximdva 3274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → (∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾_t 𝑠) ∈ Comp) → ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp))))
67	34, 66	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
68		rexcom 3355	. . . . . . 7 ⊢ (∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)) ↔ ∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
69		r19.42v 3350	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
70	69	rexbii 3247	. . . . . . 7 ⊢ (∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)) ↔ ∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
71	68, 70	bitri 277	. . . . . 6 ⊢ (∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)) ↔ ∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
72	67, 71	sylib 220	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → ∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
73	72	ralrimiva 3182	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵 “ 𝐾)∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
74	15	restuni 21770	. . . . . 6 ⊢ ((𝑆 ∈ Top ∧ (𝐵 “ 𝐾) ⊆ ∪ 𝑆) → (𝐵 “ 𝐾) = ∪ (𝑆 ↾_t (𝐵 “ 𝐾)))
75	36, 25, 74	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐵 “ 𝐾) = ∪ (𝑆 ↾_t (𝐵 “ 𝐾)))
76	75	raleqdv 3415	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (𝐵 “ 𝐾)∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)) ↔ ∀𝑥 ∈ ∪ (𝑆 ↾_t (𝐵 “ 𝐾))∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp))))
77	73, 76	mpbid 234	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝑆 ↾_t (𝐵 “ 𝐾))∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp)))
78		eqid 2821	. . . 4 ⊢ ∪ (𝑆 ↾_t (𝐵 “ 𝐾)) = ∪ (𝑆 ↾_t (𝐵 “ 𝐾))
79		fveq2 6670	. . . . . 6 ⊢ (𝑠 = (𝑘‘𝑦) → ((int‘𝑆)‘𝑠) = ((int‘𝑆)‘(𝑘‘𝑦)))
80	79	sseq2d 3999	. . . . 5 ⊢ (𝑠 = (𝑘‘𝑦) → (𝑦 ⊆ ((int‘𝑆)‘𝑠) ↔ 𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦))))
81		oveq2 7164	. . . . . 6 ⊢ (𝑠 = (𝑘‘𝑦) → (𝑆 ↾_t 𝑠) = (𝑆 ↾_t (𝑘‘𝑦)))
82	81	eleq1d 2897	. . . . 5 ⊢ (𝑠 = (𝑘‘𝑦) → ((𝑆 ↾_t 𝑠) ∈ Comp ↔ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp))
83	80, 82	anbi12d 632	. . . 4 ⊢ (𝑠 = (𝑘‘𝑦) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp) ↔ (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))
84	78, 83	cmpcovf 21999	. . 3 ⊢ (((𝑆 ↾_t (𝐵 “ 𝐾)) ∈ Comp ∧ ∀𝑥 ∈ ∪ (𝑆 ↾_t (𝐵 “ 𝐾))∃𝑦 ∈ (𝑆 ↾_t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (^◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾_t 𝑠) ∈ Comp))) → ∃𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)(∪ (𝑆 ↾_t (𝐵 “ 𝐾)) = ∪ 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp))))
85	4, 77, 84	syl2anc 586	. 2 ⊢ (𝜑 → ∃𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)(∪ (𝑆 ↾_t (𝐵 “ 𝐾)) = ∪ 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp))))
86	75	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) → (𝐵 “ 𝐾) = ∪ (𝑆 ↾_t (𝐵 “ 𝐾)))
87	86	eqeq1d 2823	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) → ((𝐵 “ 𝐾) = ∪ 𝑤 ↔ ∪ (𝑆 ↾_t (𝐵 “ 𝐾)) = ∪ 𝑤))
88	87	biimpar 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ∪ (𝑆 ↾_t (𝐵 “ 𝐾)) = ∪ 𝑤) → (𝐵 “ 𝐾) = ∪ 𝑤)
89	36	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝑆 ∈ Top)
90		cntop2 21849	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
91	7, 90	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ Top)
92	91	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝑇 ∈ Top)
93		xkotop 22196	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ↑_ko 𝑆) ∈ Top)
94	89, 92, 93	syl2anc 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝑇 ↑_ko 𝑆) ∈ Top)
95		cntop1 21848	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
96	1, 95	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Top)
97	96	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝑅 ∈ Top)
98		xkotop 22196	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑_ko 𝑅) ∈ Top)
99	97, 89, 98	syl2anc 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝑆 ↑_ko 𝑅) ∈ Top)
100		simprrl 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉))
101	100	frnd 6521	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ 𝒫 (^◡𝐴 “ 𝑉))
102		sspwuni 5022	. . . . . . . . . . . 12 ⊢ (ran 𝑘 ⊆ 𝒫 (^◡𝐴 “ 𝑉) ↔ ∪ ran 𝑘 ⊆ (^◡𝐴 “ 𝑉))
103	101, 102	sylib 220	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ ran 𝑘 ⊆ (^◡𝐴 “ 𝑉))
104	10	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (^◡𝐴 “ 𝑉) ∈ 𝑆)
105	104, 50	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (^◡𝐴 “ 𝑉) ⊆ ∪ 𝑆)
106	103, 105	sstrd 3977	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ ran 𝑘 ⊆ ∪ 𝑆)
107		ffn 6514	. . . . . . . . . . . . 13 ⊢ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) → 𝑘 Fn 𝑤)
108		fniunfv 7006	. . . . . . . . . . . . 13 ⊢ (𝑘 Fn 𝑤 → ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) = ∪ ran 𝑘)
109	100, 107, 108	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) = ∪ ran 𝑘)
110	109	oveq2d 7172	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝑆 ↾_t ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) = (𝑆 ↾_t ∪ ran 𝑘))
111		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin))
112	111	elin2d 4176	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝑤 ∈ Fin)
113		simprrr 780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp))
114		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp) → (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)
115	114	ralimi 3160	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp) → ∀𝑦 ∈ 𝑤 (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)
116	113, 115	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)
117	15	fiuncmp 22012	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Top ∧ 𝑤 ∈ Fin ∧ ∀𝑦 ∈ 𝑤 (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp) → (𝑆 ↾_t ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) ∈ Comp)
118	89, 112, 116, 117	syl3anc 1367	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝑆 ↾_t ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) ∈ Comp)
119	110, 118	eqeltrrd 2914	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝑆 ↾_t ∪ ran 𝑘) ∈ Comp)
120	8	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝑉 ∈ 𝑇)
121	15, 89, 92, 106, 119, 120	xkoopn 22197	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} ∈ (𝑇 ↑_ko 𝑆))
122		xkococn.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ⊆ ∪ 𝑅)
123	122	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝐾 ⊆ ∪ 𝑅)
124	2	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝑅 ↾_t 𝐾) ∈ Comp)
125	109, 106	eqsstrd 4005	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆)
126		iunss 4969	. . . . . . . . . . . . 13 ⊢ (∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆 ↔ ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆)
127	125, 126	sylib 220	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆)
128	15	ntropn 21657	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Top ∧ (𝑘‘𝑦) ⊆ ∪ 𝑆) → ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆)
129	128	ex 415	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Top → ((𝑘‘𝑦) ⊆ ∪ 𝑆 → ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆))
130	129	ralimdv 3178	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Top → (∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆 → ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆))
131	89, 127, 130	sylc 65	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆)
132		iunopn 21506	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Top ∧ ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆)
133	89, 131, 132	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆)
134	21, 97, 89, 123, 124, 133	xkoopn 22197	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ∈ (𝑆 ↑_ko 𝑅))
135		txopn 22210	. . . . . . . . 9 ⊢ ((((𝑇 ↑_ko 𝑆) ∈ Top ∧ (𝑆 ↑_ko 𝑅) ∈ Top) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} ∈ (𝑇 ↑_ko 𝑆) ∧ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ∈ (𝑆 ↑_ko 𝑅))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅)))
136	94, 99, 121, 134, 135	syl22anc 836	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅)))
137		imaeq1 5924	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 “ ∪ ran 𝑘) = (𝐴 “ ∪ ran 𝑘))
138	137	sseq1d 3998	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎 “ ∪ ran 𝑘) ⊆ 𝑉 ↔ (𝐴 “ ∪ ran 𝑘) ⊆ 𝑉))
139	7	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝐴 ∈ (𝑆 Cn 𝑇))
140		imaiun 7004	. . . . . . . . . . . 12 ⊢ (𝐴 “ ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) = ∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦))
141	109	imaeq2d 5929	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝐴 “ ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) = (𝐴 “ ∪ ran 𝑘))
142	140, 141	syl5eqr 2870	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) = (𝐴 “ ∪ ran 𝑘))
143	109, 103	eqsstrd 4005	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (^◡𝐴 “ 𝑉))
144	19	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → Fun 𝐴)
145	100, 107	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝑘 Fn 𝑤)
146	27	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → dom 𝐴 = ∪ 𝑆)
147	106, 146	sseqtrrd 4008	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ ran 𝑘 ⊆ dom 𝐴)
148		simpl1 1187	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦 ∈ 𝑤) → Fun 𝐴)
149	108	3ad2ant2 1130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) = ∪ ran 𝑘)
150		simp3 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → ∪ ran 𝑘 ⊆ dom 𝐴)
151	149, 150	eqsstrd 4005	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ dom 𝐴)
152		iunss 4969	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ dom 𝐴 ↔ ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ dom 𝐴)
153	151, 152	sylib 220	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ dom 𝐴)
154	153	r19.21bi 3208	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦 ∈ 𝑤) → (𝑘‘𝑦) ⊆ dom 𝐴)
155		funimass3 6824	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐴 ∧ (𝑘‘𝑦) ⊆ dom 𝐴) → ((𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ (𝑘‘𝑦) ⊆ (^◡𝐴 “ 𝑉)))
156	148, 154, 155	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦 ∈ 𝑤) → ((𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ (𝑘‘𝑦) ⊆ (^◡𝐴 “ 𝑉)))
157	156	ralbidva 3196	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → (∀𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (^◡𝐴 “ 𝑉)))
158		iunss 4969	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ ∀𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉)
159		iunss 4969	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (^◡𝐴 “ 𝑉) ↔ ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (^◡𝐴 “ 𝑉))
160	157, 158, 159	3bitr4g 316	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → (∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (^◡𝐴 “ 𝑉)))
161	144, 145, 147, 160	syl3anc 1367	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (^◡𝐴 “ 𝑉)))
162	143, 161	mpbird 259	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉)
163	142, 162	eqsstrrd 4006	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝐴 “ ∪ ran 𝑘) ⊆ 𝑉)
164	138, 139, 163	elrabd 3682	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝐴 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉})
165		imaeq1 5924	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑏 “ 𝐾) = (𝐵 “ 𝐾))
166	165	sseq1d 3998	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ↔ (𝐵 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))
167	1	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝐵 ∈ (𝑅 Cn 𝑆))
168		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝐵 “ 𝐾) = ∪ 𝑤)
169		uniiun 4982	. . . . . . . . . . . 12 ⊢ ∪ 𝑤 = ∪ 𝑦 ∈ 𝑤 𝑦
170	168, 169	syl6eq 2872	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝐵 “ 𝐾) = ∪ 𝑦 ∈ 𝑤 𝑦)
171		simpl 485	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp) → 𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)))
172	171	ralimi 3160	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp) → ∀𝑦 ∈ 𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)))
173		ss2iun 4937	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) → ∪ 𝑦 ∈ 𝑤 𝑦 ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)))
174	113, 172, 173	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 𝑦 ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)))
175	170, 174	eqsstrd 4005	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝐵 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)))
176	166, 167, 175	elrabd 3682	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → 𝐵 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})
177	164, 176	opelxpd 5593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ⟨𝐴, 𝐵⟩ ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}))
178		imaeq1 5924	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑢 → (𝑎 “ ∪ ran 𝑘) = (𝑢 “ ∪ ran 𝑘))
179	178	sseq1d 3998	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑢 → ((𝑎 “ ∪ ran 𝑘) ⊆ 𝑉 ↔ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉))
180	179	elrab 3680	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} ↔ (𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉))
181		imaeq1 5924	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑣 → (𝑏 “ 𝐾) = (𝑣 “ 𝐾))
182	181	sseq1d 3998	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑣 → ((𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ↔ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))
183	182	elrab 3680	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ↔ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))
184	180, 183	anbi12i 628	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} ∧ 𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ↔ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)))))
185		simprll 777	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → 𝑢 ∈ (𝑆 Cn 𝑇))
186		simprrl 779	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → 𝑣 ∈ (𝑅 Cn 𝑆))
187		coeq1 5728	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑢 → (𝑓 ∘ 𝑔) = (𝑢 ∘ 𝑔))
188		coeq2 5729	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑣 → (𝑢 ∘ 𝑔) = (𝑢 ∘ 𝑣))
189		xkococn.1	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓 ∘ 𝑔))
190		vex 3497	. . . . . . . . . . . . . . . 16 ⊢ 𝑢 ∈ V
191		vex 3497	. . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V
192	190, 191	coex 7635	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∘ 𝑣) ∈ V
193	187, 188, 189, 192	ovmpo 7310	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ 𝑣 ∈ (𝑅 Cn 𝑆)) → (𝑢𝐹𝑣) = (𝑢 ∘ 𝑣))
194	185, 186, 193	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢𝐹𝑣) = (𝑢 ∘ 𝑣))
195		imaeq1 5924	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑢 ∘ 𝑣) → (ℎ “ 𝐾) = ((𝑢 ∘ 𝑣) “ 𝐾))
196	195	sseq1d 3998	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑢 ∘ 𝑣) → ((ℎ “ 𝐾) ⊆ 𝑉 ↔ ((𝑢 ∘ 𝑣) “ 𝐾) ⊆ 𝑉))
197		cnco 21874	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ (𝑅 Cn 𝑆) ∧ 𝑢 ∈ (𝑆 Cn 𝑇)) → (𝑢 ∘ 𝑣) ∈ (𝑅 Cn 𝑇))
198	186, 185, 197	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 ∘ 𝑣) ∈ (𝑅 Cn 𝑇))
199		imaco 6104	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∘ 𝑣) “ 𝐾) = (𝑢 “ (𝑣 “ 𝐾))
200		simprrr 780	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)))
201	15	ntrss2 21665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ Top ∧ (𝑘‘𝑦) ⊆ ∪ 𝑆) → ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦))
202	201	ex 415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑆 ∈ Top → ((𝑘‘𝑦) ⊆ ∪ 𝑆 → ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦)))
203	202	ralimdv 3178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑆 ∈ Top → (∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆 → ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦)))
204	89, 127, 203	sylc 65	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦))
205		ss2iun 4937	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦))
206	204, 205	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦))
207	206, 109	sseqtrd 4007	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ ∪ ran 𝑘)
208	207	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ ∪ ran 𝑘)
209	200, 208	sstrd 3977	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑣 “ 𝐾) ⊆ ∪ ran 𝑘)
210		imass2 5965	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 “ 𝐾) ⊆ ∪ ran 𝑘 → (𝑢 “ (𝑣 “ 𝐾)) ⊆ (𝑢 “ ∪ ran 𝑘))
211	209, 210	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 “ (𝑣 “ 𝐾)) ⊆ (𝑢 “ ∪ ran 𝑘))
212		simprlr 778	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉)
213	211, 212	sstrd 3977	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 “ (𝑣 “ 𝐾)) ⊆ 𝑉)
214	199, 213	eqsstrid 4015	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → ((𝑢 ∘ 𝑣) “ 𝐾) ⊆ 𝑉)
215	196, 198, 214	elrabd 3682	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 ∘ 𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})
216	194, 215	eqeltrd 2913	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran 𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})
217	184, 216	sylan2b 595	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) ∧ (𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} ∧ 𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) → (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})
218	217	ralrimivva 3191	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉}∀𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})
219	189	mpofun 7276	. . . . . . . . . . 11 ⊢ Fun 𝐹
220		ssrab2 4056	. . . . . . . . . . . . 13 ⊢ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} ⊆ (𝑆 Cn 𝑇)
221		ssrab2 4056	. . . . . . . . . . . . 13 ⊢ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ⊆ (𝑅 Cn 𝑆)
222		xpss12 5570	. . . . . . . . . . . . 13 ⊢ (({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} ⊆ (𝑆 Cn 𝑇) ∧ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ⊆ (𝑅 Cn 𝑆)) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)))
223	220, 221, 222	mp2an 690	. . . . . . . . . . . 12 ⊢ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))
224		vex 3497	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
225		vex 3497	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
226	224, 225	coex 7635	. . . . . . . . . . . . 13 ⊢ (𝑓 ∘ 𝑔) ∈ V
227	189, 226	dmmpo 7769	. . . . . . . . . . . 12 ⊢ dom 𝐹 = ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))
228	223, 227	sseqtrri 4004	. . . . . . . . . . 11 ⊢ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ dom 𝐹
229		funimassov 7325	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ dom 𝐹) → ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉} ↔ ∀𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉}∀𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))
230	219, 228, 229	mp2an 690	. . . . . . . . . 10 ⊢ ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉} ↔ ∀𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉}∀𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})
231	218, 230	sylibr 236	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → (𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})
232		funimass3 6824	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ dom 𝐹) → ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉} ↔ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))
233	219, 228, 232	mp2an 690	. . . . . . . . 9 ⊢ ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉} ↔ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))
234	231, 233	sylib 220	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))
235		eleq2 2901	. . . . . . . . . 10 ⊢ (𝑧 = ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) → (⟨𝐴, 𝐵⟩ ∈ 𝑧 ↔ ⟨𝐴, 𝐵⟩ ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})))
236		sseq1 3992	. . . . . . . . . 10 ⊢ (𝑧 = ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) → (𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}) ↔ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))
237	235, 236	anbi12d 632	. . . . . . . . 9 ⊢ (𝑧 = ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) → ((⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})) ↔ (⟨𝐴, 𝐵⟩ ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))))
238	237	rspcev 3623	. . . . . . . 8 ⊢ ((({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅)) ∧ (⟨𝐴, 𝐵⟩ ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran 𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))) → ∃𝑧 ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))
239	136, 177, 234, 238	syl12anc 834	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)))) → ∃𝑧 ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))
240	239	expr 459	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ (𝐵 “ 𝐾) = ∪ 𝑤) → ((𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))))
241	240	exlimdv 1934	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ (𝐵 “ 𝐾) = ∪ 𝑤) → (∃𝑘(𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))))
242	88, 241	syldan 593	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ∪ (𝑆 ↾_t (𝐵 “ 𝐾)) = ∪ 𝑤) → (∃𝑘(𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))))
243	242	expimpd 456	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)) → ((∪ (𝑆 ↾_t (𝐵 “ 𝐾)) = ∪ 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp))) → ∃𝑧 ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))))
244	243	rexlimdva 3284	. 2 ⊢ (𝜑 → (∃𝑤 ∈ (𝒫 (𝑆 ↾_t (𝐵 “ 𝐾)) ∩ Fin)(∪ (𝑆 ↾_t (𝐵 “ 𝐾)) = ∪ 𝑤 ∧ ∃𝑘(𝑘:𝑤⟶𝒫 (^◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾_t (𝑘‘𝑦)) ∈ Comp))) → ∃𝑧 ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))))
245	85, 244	mpd 15	1 ⊢ (𝜑 → ∃𝑧 ∈ ((𝑇 ↑_ko 𝑆) ×_t (𝑆 ↑_ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (^◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ⟨cop 4573 ∪ cuni 4838 ∪ ciun 4919 × cxp 5553 ^◡ccnv 5554 dom cdm 5555 ran crn 5556 “ cima 5558 ∘ ccom 5559 Fun wfun 6349 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Fincfn 8509 ↾_t crest 16694 Topctop 21501 intcnt 21625 Cn ccn 21832 Compccmp 21994 𝑛-Locally cnlly 22073 ×_t ctx 22168 ↑_ko cxko 22169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-fin 8513 df-fi 8875 df-rest 16696 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 df-ntr 21628 df-nei 21706 df-cn 21835 df-cmp 21995 df-nlly 22075 df-tx 22170 df-xko 22171

This theorem is referenced by: xkococn 22268

W3C validator