Theorem xkoptsub 23811

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 23063	. . . . . . . 8 ⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅)
4	3	adantr 485	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑋 ∈ 𝑅)
5		fconstg 6765	. . . . . . . . 9 ⊢ (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶{𝑆})
6	5	adantl 486	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶{𝑆})
7	6	ffnd 6706	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}) Fn 𝑋)
8		eqid 2763	. . . . . . . 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 23727	. . . . . . 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 595	. . . . . 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 489	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
12	11	snssd 4752	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑆} ⊆ Top)
13	6, 12	fssd 6723	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶Top)
14		eqid 2763	. . . . . . . . . 10 ⊢ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)
15	8, 14	ptbasfi 23738	. . . . . . . . 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 595	. . . . . . . 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 7200	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Top ∧ 𝑛 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆)
18	17	adantll 726	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆)
19	18	unieqd 4885	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛 ∈ 𝑋) → ∪ ((𝑋 × {𝑆})‘𝑛) = ∪ 𝑆)
20	19	ixpeq2dva 8906	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = X𝑛 ∈ 𝑋 ∪ 𝑆)
21		eqid 2763	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑆 = ∪ 𝑆
22	21	topopn 23063	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
23		ixpconstg 8900	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆) → X𝑛 ∈ 𝑋 ∪ 𝑆 = (∪ 𝑆 ↑m 𝑋))
24	3, 22, 23	syl2an 607	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈ 𝑋 ∪ 𝑆 = (∪ 𝑆 ↑m 𝑋))
25	20, 24	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = (∪ 𝑆 ↑m 𝑋))
26	25	sneqd 4601	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)} = {(∪ 𝑆 ↑m 𝑋)})
27		eqid 2763	. . . . . . . . . . . 12 ⊢ 𝑋 = 𝑋
28		fvconst2g 7200	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Top ∧ 𝑘 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆)
29	28	adantll 726	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆)
30	25	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = (∪ 𝑆 ↑m 𝑋))
31	30	mpteq1d 5201	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → (𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) = (𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)))
32	31	cnveqd 5861	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)))
33	32	imaeq1d 6061	. . . . . . . . . . . . . . 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 3161	. . . . . . . . . . . . . 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 520	. . . . . . . . . . . . 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 3157	. . . . . . . . . . . 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 7482	. . . . . . . . . . . 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 598	. . . . . . . . . . 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 5928	. . . . . . . . . 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 4123	. . . . . . . . 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 6885	. . . . . . . 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 2798	. . . . . . 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 6885	. . . . . 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 2798	. . . . 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 2810	. . . 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 7425	. . 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 17480	. . . . 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 6884	. . . 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 6894	. . . . 5 ⊢ (fi‘({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ∈ V
50		ovex 7443	. . . . 5 ⊢ (𝑅 Cn 𝑆) ∈ V
51		tgrest 23316	. . . . 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 704	. . . 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 2786	. . 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 2816	. 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 23745	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ Top)
56		snex 5410	. . . . . 6 ⊢ {(∪ 𝑆 ↑m 𝑋)} ∈ V
57		mpoexga 8070	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
58	3, 57	sylan 591	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
59		rnexg 7895	. . . . . . 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 18	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
61		unexg 7741	. . . . . 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 598	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ({(∪ 𝑆 ↑m 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
63		restval 17474	. . . . 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 597	. . . 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 4107	. . . . . . 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 23403	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥:𝑋⟶∪ 𝑆)
67		elmapg 8832	. . . . . . . . . . . . . . 15 ⊢ ((∪ 𝑆 ∈ 𝑆 ∧ 𝑋 ∈ 𝑅) → (𝑥 ∈ (∪ 𝑆 ↑m 𝑋) ↔ 𝑥:𝑋⟶∪ 𝑆))
68	22, 3, 67	syl2anr 608	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ (∪ 𝑆 ↑m 𝑋) ↔ 𝑥:𝑋⟶∪ 𝑆))
69	66, 68	imbitrrid 249	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥 ∈ (∪ 𝑆 ↑m 𝑋)))
70	69	ssrdv 3943	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
71	70	adantr 485	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
72		elsni 4606	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)} → 𝑥 = (∪ 𝑆 ↑m 𝑋))
73	72	adantl 486	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → 𝑥 = (∪ 𝑆 ↑m 𝑋))
74	71, 73	sseqtrrd 3974	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑅 Cn 𝑆) ⊆ 𝑥)
75		sseqin2 4176	. . . . . . . . . 10 ⊢ ((𝑅 Cn 𝑆) ⊆ 𝑥 ↔ (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆))
76	74, 75	sylib 221	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆))
77		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
78	77	xkouni 23756	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ (𝑆 ↑ko 𝑅))
79		eqid 2763	. . . . . . . . . . . . 13 ⊢ ∪ (𝑆 ↑ko 𝑅) = ∪ (𝑆 ↑ko 𝑅)
80	79	topopn 23063	. . . . . . . . . . . 12 ⊢ ((𝑆 ↑ko 𝑅) ∈ Top → ∪ (𝑆 ↑ko 𝑅) ∈ (𝑆 ↑ko 𝑅))
81	55, 80	syl 18	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ (𝑆 ↑ko 𝑅) ∈ (𝑆 ↑ko 𝑅))
82	78, 81	eqeltrd 2863	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅))
83	82	adantr 485	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑅 Cn 𝑆) ∈ (𝑆 ↑ko 𝑅))
84	76, 83	eqeltrd 2863	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑m 𝑋)}) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))
85		eqid 2763	. . . . . . . . . . 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 7543	. . . . . . . . . 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	eqabri 2905	. . . . . . . . 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 6231	. . . . . . . . . . . . . . 15 ⊢ (◡((𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) “ 𝑢) = ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆))
89	70	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
90	89	resmptd 6042	. . . . . . . . . . . . . . . . 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 5861	. . . . . . . . . . . . . . . 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 6061	. . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤‘𝑘) ∈ V
95	94	rgenw 3083	. . . . . . . . . . . . . . . . . . 19 ⊢ ∀𝑤 ∈ (𝑅 Cn 𝑆)(𝑤‘𝑘) ∈ V
96		eqid 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) = (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))
97	96	fnmpt 6675	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑤 ∈ (𝑅 Cn 𝑆)(𝑤‘𝑘) ∈ V → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆))
98	95, 97	mp1i 14	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆))
99		elpreima 7053	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑓 → (𝑤‘𝑘) = (𝑓‘𝑘))
102		fvex 6894	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓‘𝑘) ∈ V
103	101, 96, 102	fvmpt 6989	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) = (𝑓‘𝑘))
104	103	adantl 486	. . . . . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . 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 4750	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑘) ∈ 𝑢 ↔ {(𝑓‘𝑘)} ⊆ 𝑢)
107	89	sselda 3937	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 ∈ (∪ 𝑆 ↑m 𝑋))
108		elmapi 8842	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ (∪ 𝑆 ↑m 𝑋) → 𝑓:𝑋⟶∪ 𝑆)
109		ffn 6705	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝑋⟶∪ 𝑆 → 𝑓 Fn 𝑋)
110	107, 108, 109	3syl 19	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 Fn 𝑋)
111		simplrl 788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑘 ∈ 𝑋)
112		fnsnfv 6960	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 Fn 𝑋 ∧ 𝑘 ∈ 𝑋) → {(𝑓‘𝑘)} = (𝑓 “ {𝑘}))
113	110, 111, 112	syl2anc 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → {(𝑓‘𝑘)} = (𝑓 “ {𝑘}))
114	113	sseq1d 3968	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ({(𝑓‘𝑘)} ⊆ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
115	106, 114	bitrid 286	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ((𝑓‘𝑘) ∈ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
116	105, 115	bitrd 282	. . . . . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . . . 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	eqabdv 2896	. . . . . . . . . . . . . . 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 3417	. . . . . . . . . . . . . . 15 ⊢ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} = {𝑓 ∣ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)}
121	119, 120	eqtr4di 2816	. . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . 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 778	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑅 ∈ Top)
124	11	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ∈ Top)
125		simprl 782	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑘 ∈ 𝑋)
126	125	snssd 4752	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → {𝑘} ⊆ 𝑋)
127	2	toptopon 23074	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
128	123, 127	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑅 ∈ (TopOn‘𝑋))
129		restsn2 23328	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝑋) → (𝑅 ↾t {𝑘}) = 𝒫 {𝑘})
130	128, 125, 129	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 ↾t {𝑘}) = 𝒫 {𝑘})
131		snfi 9036	. . . . . . . . . . . . . . . 16 ⊢ {𝑘} ∈ Fin
132		discmp 23555	. . . . . . . . . . . . . . . 16 ⊢ ({𝑘} ∈ Fin ↔ 𝒫 {𝑘} ∈ Comp)
133	131, 132	mpbi 233	. . . . . . . . . . . . . . 15 ⊢ 𝒫 {𝑘} ∈ Comp
134	130, 133	eqeltrdi 2871	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 ↾t {𝑘}) ∈ Comp)
135		simprr 784	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆)
136	2, 123, 124, 126, 134, 135	xkoopn 23746	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} ∈ (𝑆 ↑ko 𝑅))
137	122, 136	eqeltrd 2863	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((◡(𝑤 ∈ (∪ 𝑆 ↑m 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ↑ko 𝑅))
138		ineq1 4166	. . . . . . . . . . . . 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 2848	. . . . . . . . . . . 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 250	. . . . . . . . . . 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 3222	. . . . . . . . . 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 411	. . . . . . . . 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 605	. . . . . . . 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 972	. . . . . . 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 605	. . . . . 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 7110	. . . . 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 6714	. . . 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 3971	. . 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 23148	. . 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 595	. 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 3971	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽 ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872   ↦ cmpt 5192   × cxp 5659  ^◡ccnv 5660  ran crn 5662   ↾ cres 5663   “ cima 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ↑_m cmap 8820  Xcixp 8891  Fincfn 8939  ficfi 9366   ↾_t crest 17468  topGenctg 17485  ∏_tcpt 17486  Topctop 23050  TopOnctopon 23067   Cn ccn 23381  Compccmp 23543   ↑_ko cxko 23718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-1o 8449  df-2o 8450  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9367  df-rest 17470  df-topgen 17491  df-pt 17492  df-top 23051  df-topon 23068  df-bases 23103  df-cn 23384  df-cmp 23544  df-xko 23720

This theorem is referenced by:  xkopt  23812  xkopjcn  23813