Theorem xkoptsub 23005

Description: The compact-open topology is finer than the product topology restricted to continuous functions. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypotheses

Ref	Expression
xkoptsub.x	⊢ 𝑋 = ∪ 𝑅
xkoptsub.j	⊢ 𝐽 = (∏t‘(𝑋 × {𝑆}))

Assertion

Ref	Expression
xkoptsub	⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽 ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))

Proof of Theorem xkoptsub

Dummy variables 𝑓 𝑔 𝑘 𝑛 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xkoptsub.j	. . . . 5 ⊢ 𝐽 = (∏t‘(𝑋 × {𝑆}))
2		xkoptsub.x	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝑅
3	2	topopn 22255	. . . . . . . 8 ⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅)
4	3	adantr 481	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑋 ∈ 𝑅)
5		fconstg 6729	. . . . . . . . 9 ⊢ (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶{𝑆})
6	5	adantl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶{𝑆})
7	6	ffnd 6669	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}) Fn 𝑋)
8		eqid 2736	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))}
9	8	ptval 22921	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}) Fn 𝑋) → (∏t‘(𝑋 × {𝑆})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))}))
10	4, 7, 9	syl2anc 584	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∏t‘(𝑋 × {𝑆})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))}))
11		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
12	11	snssd 4769	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑆} ⊆ Top)
13	6, 12	fssd 6686	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶Top)
14		eqid 2736	. . . . . . . . . 10 ⊢ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)
15	8, 14	ptbasfi 22932	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))} = (fi‘({X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)))))
16	4, 13, 15	syl2anc 584	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))} = (fi‘({X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)))))
17		fvconst2g 7151	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Top ∧ 𝑛 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆)
18	17	adantll 712	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆)
19	18	unieqd 4879	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛 ∈ 𝑋) → ∪ ((𝑋 × {𝑆})‘𝑛) = ∪ 𝑆)
20	19	ixpeq2dva 8850	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = X𝑛 ∈ 𝑋 ∪ 𝑆)
21		eqid 2736	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑆 = ∪ 𝑆
22	21	topopn 22255	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
23		ixpconstg 8844	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆) → X𝑛 ∈ 𝑋 ∪ 𝑆 = (∪ 𝑆 ↑m 𝑋))
24	3, 22, 23	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈ 𝑋 ∪ 𝑆 = (∪ 𝑆 ↑m 𝑋))
25	20, 24	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = (∪ 𝑆 ↑m 𝑋))
26	25	sneqd 4598	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)} = {(∪ 𝑆 ↑m 𝑋)})
27		eqid 2736	. . . . . . . . . . . 12 ⊢ 𝑋 = 𝑋
28		fvconst2g 7151	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Top ∧ 𝑘 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆)
29	28	adantll 712	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆)
30	25	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = (∪ 𝑆 ↑m 𝑋))
31	30	mpteq1d 5200	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → (𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) = (𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)))
32	31	cnveqd 5831	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)))
33	32	imaeq1d 6012	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))
34	33	ralrimivw 3147	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))
35	29, 34	jca 512	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → (((𝑋 × {𝑆})‘𝑘) = 𝑆 ∧ ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))
36	35	ralrimiva 3143	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∀𝑘 ∈ 𝑋 (((𝑋 × {𝑆})‘𝑘) = 𝑆 ∧ ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))
37		mpoeq123 7429	. . . . . . . . . . . 12 ⊢ ((𝑋 = 𝑋 ∧ ∀𝑘 ∈ 𝑋 (((𝑋 × {𝑆})‘𝑘) = 𝑆 ∧ ∀𝑢 ∈ ((𝑋 × {𝑆})‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) → (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))
38	27, 36, 37	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))
39	38	rneqd 5893	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))
40	26, 39	uneq12d 4124	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ({X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))) = ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))
41	40	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (fi‘({X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ ((𝑋 × {𝑆})‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)))) = (fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))))
42	16, 41	eqtrd 2776	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))} = (fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))))
43	42	fveq2d 6846	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝑔‘𝑦) ∈ ((𝑋 × {𝑆})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝑋 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝑋 × {𝑆})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝑋 (𝑔‘𝑦))}) = (topGen‘(fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))))
44	10, 43	eqtrd 2776	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∏t‘(𝑋 × {𝑆})) = (topGen‘(fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))))
45	1, 44	eqtrid 2788	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 = (topGen‘(fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))))
46	45	oveq1d 7372	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽 ↾t (𝑅 Cn 𝑆)) = ((topGen‘(fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆)))
47		firest 17314	. . . . 5 ⊢ (fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆))) = ((fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆))
48	47	fveq2i 6845	. . . 4 ⊢ (topGen‘(fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) = (topGen‘((fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆)))
49		fvex 6855	. . . . 5 ⊢ (fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ∈ V
50		ovex 7390	. . . . 5 ⊢ (𝑅 Cn 𝑆) ∈ V
51		tgrest 22510	. . . . 5 ⊢ (((fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ∈ V ∧ (𝑅 Cn 𝑆) ∈ V) → (topGen‘((fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆))) = ((topGen‘(fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆)))
52	49, 50, 51	mp2an 690	. . . 4 ⊢ (topGen‘((fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ↾t (𝑅 Cn 𝑆))) = ((topGen‘(fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆))
53	48, 52	eqtri 2764	. . 3 ⊢ (topGen‘(fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) = ((topGen‘(fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))) ↾t (𝑅 Cn 𝑆))
54	46, 53	eqtr4di 2794	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽 ↾t (𝑅 Cn 𝑆)) = (topGen‘(fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))))
55		xkotop 22939	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ Top)
56		snex 5388	. . . . . 6 ⊢ {(∪ 𝑆 ↑m 𝑋)} ∈ V
57		mpoexga 8010	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
58	3, 57	sylan 580	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
59		rnexg 7841	. . . . . . 7 ⊢ ((𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V → ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
60	58, 59	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
61		unexg 7683	. . . . . 6 ⊢ (({(∪ 𝑆 ↑m 𝑋)} ∈ V ∧ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V) → ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
62	56, 60, 61	sylancr 587	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
63		restval 17308	. . . . 5 ⊢ ((({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V ∧ (𝑅 Cn 𝑆) ∈ V) → (({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) = ran (𝑥 ∈ ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))))
64	62, 50, 63	sylancl 586	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) = ran (𝑥 ∈ ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))))
65		elun 4108	. . . . . . 7 ⊢ (𝑥 ∈ ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↔ (𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)} ∨ 𝑥 ∈ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))
66	2, 21	cnf 22597	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥:𝑋⟶∪ 𝑆)
67		elmapg 8778	. . . . . . . . . . . . . . 15 ⊢ ((∪ 𝑆 ∈ 𝑆 ∧ 𝑋 ∈ 𝑅) → (𝑥 ∈ (∪ 𝑆 ↑m 𝑋) ↔ 𝑥:𝑋⟶∪ 𝑆))
68	22, 3, 67	syl2anr 597	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ (∪ 𝑆 ↑m 𝑋) ↔ 𝑥:𝑋⟶∪ 𝑆))
69	66, 68	syl5ibr 245	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥 ∈ (∪ 𝑆 ↑m 𝑋)))
70	69	ssrdv 3950	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
71	70	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
72		elsni 4603	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)} → 𝑥 = (∪ 𝑆 ↑m 𝑋))
73	72	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → 𝑥 = (∪ 𝑆 ↑m 𝑋))
74	71, 73	sseqtrrd 3985	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑅 Cn 𝑆) ⊆ 𝑥)
75		sseqin2 4175	. . . . . . . . . 10 ⊢ ((𝑅 Cn 𝑆) ⊆ 𝑥 ↔ (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆))
76	74, 75	sylib 217	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆))
77		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
78	77	xkouni 22950	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ (𝑆 ↑ko 𝑅))
79		eqid 2736	. . . . . . . . . . . . 13 ⊢ ∪ (𝑆 ↑ko 𝑅) = ∪ (𝑆 ↑ko 𝑅)
80	79	topopn 22255	. . . . . . . . . . . 12 ⊢ ((𝑆 ↑ko 𝑅) ∈ Top → ∪ (𝑆 ↑ko 𝑅) ∈ (𝑆 ↑ko 𝑅))
81	55, 80	syl 17	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ (𝑆 ↑ko 𝑅) ∈ (𝑆 ↑ko 𝑅))
82	78, 81	eqeltrd 2838	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅))
83	82	adantr 481	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅))
84	76, 83	eqeltrd 2838	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))
85		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))
86	85	rnmpo 7489	. . . . . . . . . 10 ⊢ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) = {𝑥 ∣ ∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)}
87	86	eqabi 2881	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ↔ ∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))
88		cnvresima 6182	. . . . . . . . . . . . . . 15 ⊢ (◡((𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) “ 𝑢) = ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆))
89	70	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
90	89	resmptd 5994	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) = (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)))
91	90	cnveqd 5831	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ◡((𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) = ◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)))
92	91	imaeq1d 6012	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (◡((𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) “ 𝑢) = (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢))
93	88, 92	eqtr3id 2790	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) = (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢))
94		fvex 6855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤‘𝑘) ∈ V
95	94	rgenw 3068	. . . . . . . . . . . . . . . . . . 19 ⊢ ∀𝑤 ∈ (𝑅 Cn 𝑆)(𝑤‘𝑘) ∈ V
96		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) = (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))
97	96	fnmpt 6641	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑤 ∈ (𝑅 Cn 𝑆)(𝑤‘𝑘) ∈ V → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆))
98	95, 97	mp1i 13	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆))
99		elpreima 7008	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆) → (𝑓 ∈ (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢)))
100	98, 99	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑓 ∈ (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢)))
101		fveq1 6841	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑓 → (𝑤‘𝑘) = (𝑓‘𝑘))
102		fvex 6855	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓‘𝑘) ∈ V
103	101, 96, 102	fvmpt 6948	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) = (𝑓‘𝑘))
104	103	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) = (𝑓‘𝑘))
105	104	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢 ↔ (𝑓‘𝑘) ∈ 𝑢))
106	102	snss 4746	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑘) ∈ 𝑢 ↔ {(𝑓‘𝑘)} ⊆ 𝑢)
107	89	sselda 3944	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 ∈ (∪ 𝑆 ↑m 𝑋))
108		elmapi 8787	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ (∪ 𝑆 ↑m 𝑋) → 𝑓:𝑋⟶∪ 𝑆)
109		ffn 6668	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝑋⟶∪ 𝑆 → 𝑓 Fn 𝑋)
110	107, 108, 109	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 Fn 𝑋)
111		simplrl 775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑘 ∈ 𝑋)
112		fnsnfv 6920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 Fn 𝑋 ∧ 𝑘 ∈ 𝑋) → {(𝑓‘𝑘)} = (𝑓 “ {𝑘}))
113	110, 111, 112	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → {(𝑓‘𝑘)} = (𝑓 “ {𝑘}))
114	113	sseq1d 3975	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ({(𝑓‘𝑘)} ⊆ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
115	106, 114	bitrid 282	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ((𝑓‘𝑘) ∈ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
116	105, 115	bitrd 278	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
117	116	pm5.32da 579	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((𝑓 ∈ (𝑅 Cn 𝑆) ∧ ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)))
118	100, 117	bitrd 278	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑓 ∈ (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)))
119	118	abbi2dv 2871	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑓 ∣ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)})
120		df-rab 3408	. . . . . . . . . . . . . . 15 ⊢ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} = {𝑓 ∣ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)}
121	119, 120	eqtr4di 2794	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢})
122	93, 121	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢})
123		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑅 ∈ Top)
124	11	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ∈ Top)
125		simprl 769	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑘 ∈ 𝑋)
126	125	snssd 4769	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → {𝑘} ⊆ 𝑋)
127	2	toptopon 22266	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
128	123, 127	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑅 ∈ (TopOn‘𝑋))
129		restsn2 22522	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝑋) → (𝑅 ↾t {𝑘}) = 𝒫 {𝑘})
130	128, 125, 129	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 ↾t {𝑘}) = 𝒫 {𝑘})
131		snfi 8988	. . . . . . . . . . . . . . . 16 ⊢ {𝑘} ∈ Fin
132		discmp 22749	. . . . . . . . . . . . . . . 16 ⊢ ({𝑘} ∈ Fin ↔ 𝒫 {𝑘} ∈ Comp)
133	131, 132	mpbi 229	. . . . . . . . . . . . . . 15 ⊢ 𝒫 {𝑘} ∈ Comp
134	130, 133	eqeltrdi 2846	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 ↾t {𝑘}) ∈ Comp)
135		simprr 771	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆)
136	2, 123, 124, 126, 134, 135	xkoopn 22940	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} ∈ (𝑆 ↑ko 𝑅))
137	122, 136	eqeltrd 2838	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))
138		ineq1 4165	. . . . . . . . . . . . 13 ⊢ (𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) = ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)))
139	138	eleq1d 2822	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → ((𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅) ↔ ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅)))
140	137, 139	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅)))
141	140	rexlimdvva 3205	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅)))
142	141	imp 407	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))
143	87, 142	sylan2b 594	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))
144	84, 143	jaodan 956	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)} ∨ 𝑥 ∈ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))
145	65, 144	sylan2b 594	. . . . . 6 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))
146	145	fmpttd 7063	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))):({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))⟶(𝑆 ↑ko 𝑅))
147	146	frnd 6676	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑥 ∈ ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))) ⊆ (𝑆 ↑ko 𝑅))
148	64, 147	eqsstrd 3982	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
149		tgfiss 22341	. . 3 ⊢ (((𝑆 ↑ko 𝑅) ∈ Top ∧ (({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅)) → (topGen‘(fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) ⊆ (𝑆 ↑ko 𝑅))
150	55, 148, 149	syl2anc 584	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (topGen‘(fi‘(({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↾t (𝑅 Cn 𝑆)))) ⊆ (𝑆 ↑ko 𝑅))
151	54, 150	eqsstrd 3982	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽 ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  𝒫 cpw 4560  {csn 4586  ∪ cuni 4865   ↦ cmpt 5188   × cxp 5631  ^◡ccnv 5632  ran crn 5634   ↾ cres 5635   “ cima 5636   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359   ↑_m cmap 8765  Xcixp 8835  Fincfn 8883  ficfi 9346   ↾_t crest 17302  topGenctg 17319  ∏_tcpt 17320  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  Compccmp 22737   ↑_ko cxko 22912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-rest 17304  df-topgen 17325  df-pt 17326  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-cmp 22738  df-xko 22914

This theorem is referenced by:  xkopt  23006  xkopjcn  23007