Theorem xkoptsub 21457

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 20711	. . . . . . . 8 ⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅)
4	3	adantr 481	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑋 ∈ 𝑅)
5		fconstg 6092	. . . . . . . . 9 ⊢ (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶{𝑆})
6	5	adantl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶{𝑆})
7		ffn 6045	. . . . . . . 8 ⊢ ((𝑋 × {𝑆}):𝑋⟶{𝑆} → (𝑋 × {𝑆}) Fn 𝑋)
8	6, 7	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}) Fn 𝑋)
9		eqid 2622	. . . . . . . 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))}
10	9	ptval 21373	. . . . . . 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))}))
11	4, 8, 10	syl2anc 693	. . . . . 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))}))
12		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
13	12	snssd 4340	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {𝑆} ⊆ Top)
14	6, 13	fssd 6057	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × {𝑆}):𝑋⟶Top)
15		eqid 2622	. . . . . . . . . 10 ⊢ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)
16	9, 15	ptbasfi 21384	. . . . . . . . 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)))))
17	4, 14, 16	syl2anc 693	. . . . . . . 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)))))
18		fvconst2g 6467	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Top ∧ 𝑛 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆)
19	18	adantll 750	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑛) = 𝑆)
20	19	unieqd 4446	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑛 ∈ 𝑋) → ∪ ((𝑋 × {𝑆})‘𝑛) = ∪ 𝑆)
21	20	ixpeq2dva 7923	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = X𝑛 ∈ 𝑋 ∪ 𝑆)
22		eqid 2622	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑆 = ∪ 𝑆
23	22	topopn 20711	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
24		ixpconstg 7917	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆) → X𝑛 ∈ 𝑋 ∪ 𝑆 = (∪ 𝑆 ↑𝑚 𝑋))
25	3, 23, 24	syl2an 494	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈ 𝑋 ∪ 𝑆 = (∪ 𝑆 ↑𝑚 𝑋))
26	21, 25	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = (∪ 𝑆 ↑𝑚 𝑋))
27	26	sneqd 4189	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → {X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛)} = {(∪ 𝑆 ↑𝑚 𝑋)})
28		eqid 2622	. . . . . . . . . . . 12 ⊢ 𝑋 = 𝑋
29		fvconst2g 6467	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Top ∧ 𝑘 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆)
30	29	adantll 750	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝑘) = 𝑆)
31	26	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) = (∪ 𝑆 ↑𝑚 𝑋))
32	31	mpteq1d 4738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → (𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) = (𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)))
33	32	cnveqd 5298	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → ◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)))
34	33	imaeq1d 5465	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑘 ∈ 𝑋) → (◡(𝑤 ∈ X𝑛 ∈ 𝑋 ∪ ((𝑋 × {𝑆})‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))
35	34	ralrimivw 2967	. . . . . . . . . . . . . 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))
36	30, 35	jca 554	. . . . . . . . . . . . 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)))
37	36	ralrimiva 2966	. . . . . . . . . . . 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)))
38		mpt2eq123 6714	. . . . . . . . . . . 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)))
39	28, 37, 38	sylancr 695	. . . . . . . . . . 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)))
40	39	rneqd 5353	. . . . . . . . . 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)))
41	27, 40	uneq12d 3768	. . . . . . . . 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))))
42	41	fveq2d 6195	. . . . . . . 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)))))
43	17, 42	eqtrd 2656	. . . . . . 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)))))
44	43	fveq2d 6195	. . . . . 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))))))
45	11, 44	eqtrd 2656	. . . . 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))))))
46	1, 45	syl5eq 2668	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 = (topGen‘(fi‘({(∪ 𝑆 ↑𝑚 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))))))
47	46	oveq1d 6665	. . 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)))
48		firest 16093	. . . . 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))
49	48	fveq2i 6194	. . . 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)))
50		fvex 6201	. . . . 5 ⊢ (fi‘({(∪ 𝑆 ↑𝑚 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)))) ∈ V
51		ovex 6678	. . . . 5 ⊢ (𝑅 Cn 𝑆) ∈ V
52		tgrest 20963	. . . . 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)))
53	50, 51, 52	mp2an 708	. . . 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))
54	49, 53	eqtri 2644	. . 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))
55	47, 54	syl6eqr 2674	. 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)))))
56		xkotop 21391	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
57		snex 4908	. . . . . 6 ⊢ {(∪ 𝑆 ↑𝑚 𝑋)} ∈ V
58		mpt2exga 7246	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
59	3, 58	sylan 488	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
60		rnexg 7098	. . . . . . 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)
61	59, 60	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
62		unexg 6959	. . . . . 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)
63	57, 61, 62	sylancr 695	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ({(∪ 𝑆 ↑𝑚 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
64		restval 16087	. . . . 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))))
65	63, 51, 64	sylancl 694	. . . 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))))
66		elun 3753	. . . . . . 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))))
67	2, 22	cnf 21050	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥:𝑋⟶∪ 𝑆)
68		elmapg 7870	. . . . . . . . . . . . . . 15 ⊢ ((∪ 𝑆 ∈ 𝑆 ∧ 𝑋 ∈ 𝑅) → (𝑥 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↔ 𝑥:𝑋⟶∪ 𝑆))
69	23, 3, 68	syl2anr 495	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↔ 𝑥:𝑋⟶∪ 𝑆))
70	67, 69	syl5ibr 236	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑥 ∈ (𝑅 Cn 𝑆) → 𝑥 ∈ (∪ 𝑆 ↑𝑚 𝑋)))
71	70	ssrdv 3609	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑𝑚 𝑋))
72	71	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑𝑚 𝑋)}) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑𝑚 𝑋))
73		elsni 4194	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {(∪ 𝑆 ↑𝑚 𝑋)} → 𝑥 = (∪ 𝑆 ↑𝑚 𝑋))
74	73	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑𝑚 𝑋)}) → 𝑥 = (∪ 𝑆 ↑𝑚 𝑋))
75	72, 74	sseqtr4d 3642	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑𝑚 𝑋)}) → (𝑅 Cn 𝑆) ⊆ 𝑥)
76		sseqin2 3817	. . . . . . . . . 10 ⊢ ((𝑅 Cn 𝑆) ⊆ 𝑥 ↔ (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆))
77	75, 76	sylib 208	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑𝑚 𝑋)}) → (𝑥 ∩ (𝑅 Cn 𝑆)) = (𝑅 Cn 𝑆))
78		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
79	78	xkouni 21402	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ (𝑆 ^ko 𝑅))
80		eqid 2622	. . . . . . . . . . . . 13 ⊢ ∪ (𝑆 ^ko 𝑅) = ∪ (𝑆 ^ko 𝑅)
81	80	topopn 20711	. . . . . . . . . . . 12 ⊢ ((𝑆 ^ko 𝑅) ∈ Top → ∪ (𝑆 ^ko 𝑅) ∈ (𝑆 ^ko 𝑅))
82	56, 81	syl 17	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ (𝑆 ^ko 𝑅) ∈ (𝑆 ^ko 𝑅))
83	79, 82	eqeltrd 2701	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) ∈ (𝑆 ^ko 𝑅))
84	83	adantr 481	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑𝑚 𝑋)}) → (𝑅 Cn 𝑆) ∈ (𝑆 ^ko 𝑅))
85	77, 84	eqeltrd 2701	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ 𝑥 ∈ {(∪ 𝑆 ↑𝑚 𝑋)}) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ^ko 𝑅))
86		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))
87	86	rnmpt2 6770	. . . . . . . . . 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)}
88	87	abeq2i 2735	. . . . . . . . 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))
89		cnvresima 5623	. . . . . . . . . . . . . . 15 ⊢ (◡((𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) “ 𝑢) = ((◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆))
90	71	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑𝑚 𝑋))
91	90	resmptd 5452	. . . . . . . . . . . . . . . . 17 ⊢ (((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	cnveqd 5298	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ◡((𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) ↾ (𝑅 Cn 𝑆)) = ◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)))
93	92	imaeq1d 5465	. . . . . . . . . . . . . . 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))
94	89, 93	syl5eqr 2670	. . . . . . . . . . . . . 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))
95		fvex 6201	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤‘𝑘) ∈ V
96	95	rgenw 2924	. . . . . . . . . . . . . . . . . . 19 ⊢ ∀𝑤 ∈ (𝑅 Cn 𝑆)(𝑤‘𝑘) ∈ V
97		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) = (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))
98	97	fnmpt 6020	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑤 ∈ (𝑅 Cn 𝑆)(𝑤‘𝑘) ∈ V → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆))
99	96, 98	mp1i 13	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) Fn (𝑅 Cn 𝑆))
100		elpreima 6337	. . . . . . . . . . . . . . . . . 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)))
101	99, 100	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)))
102		fveq1 6190	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑓 → (𝑤‘𝑘) = (𝑓‘𝑘))
103		fvex 6201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓‘𝑘) ∈ V
104	102, 97, 103	fvmpt 6282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) = (𝑓‘𝑘))
105	104	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) = (𝑓‘𝑘))
106	105	eleq1d 2686	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢 ↔ (𝑓‘𝑘) ∈ 𝑢))
107	103	snss 4316	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑘) ∈ 𝑢 ↔ {(𝑓‘𝑘)} ⊆ 𝑢)
108	90	sselda 3603	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 ∈ (∪ 𝑆 ↑𝑚 𝑋))
109		elmapi 7879	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ (∪ 𝑆 ↑𝑚 𝑋) → 𝑓:𝑋⟶∪ 𝑆)
110		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝑋⟶∪ 𝑆 → 𝑓 Fn 𝑋)
111	108, 109, 110	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑓 Fn 𝑋)
112		simplrl 800	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → 𝑘 ∈ 𝑋)
113		fnsnfv 6258	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 Fn 𝑋 ∧ 𝑘 ∈ 𝑋) → {(𝑓‘𝑘)} = (𝑓 “ {𝑘}))
114	111, 112, 113	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → {(𝑓‘𝑘)} = (𝑓 “ {𝑘}))
115	114	sseq1d 3632	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ({(𝑓‘𝑘)} ⊆ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
116	107, 115	syl5bb 272	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → ((𝑓‘𝑘) ∈ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
117	106, 116	bitrd 268	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) ∧ 𝑓 ∈ (𝑅 Cn 𝑆)) → (((𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘))‘𝑓) ∈ 𝑢 ↔ (𝑓 “ {𝑘}) ⊆ 𝑢))
118	117	pm5.32da 673	. . . . . . . . . . . . . . . . 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)))
119	101, 118	bitrd 268	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑓 ∈ (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)))
120	119	abbi2dv 2742	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑓 ∣ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)})
121		df-rab 2921	. . . . . . . . . . . . . . 15 ⊢ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} = {𝑓 ∣ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ (𝑓 “ {𝑘}) ⊆ 𝑢)}
122	120, 121	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (◡(𝑤 ∈ (𝑅 Cn 𝑆) ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢})
123	94, 122	eqtrd 2656	. . . . . . . . . . . . 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})
124		simpll 790	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑅 ∈ Top)
125	12	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ∈ Top)
126		simprl 794	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑘 ∈ 𝑋)
127	126	snssd 4340	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → {𝑘} ⊆ 𝑋)
128	2	toptopon 20722	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
129	124, 128	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑅 ∈ (TopOn‘𝑋))
130		restsn2 20975	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝑋) → (𝑅 ↾t {𝑘}) = 𝒫 {𝑘})
131	129, 126, 130	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 ↾t {𝑘}) = 𝒫 {𝑘})
132		snfi 8038	. . . . . . . . . . . . . . . 16 ⊢ {𝑘} ∈ Fin
133		discmp 21201	. . . . . . . . . . . . . . . 16 ⊢ ({𝑘} ∈ Fin ↔ 𝒫 {𝑘} ∈ Comp)
134	132, 133	mpbi 220	. . . . . . . . . . . . . . 15 ⊢ 𝒫 {𝑘} ∈ Comp
135	131, 134	syl6eqel 2709	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → (𝑅 ↾t {𝑘}) ∈ Comp)
136		simprr 796	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆)
137	2, 124, 125, 127, 135, 136	xkoopn 21392	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ {𝑘}) ⊆ 𝑢} ∈ (𝑆 ^ko 𝑅))
138	123, 137	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑘 ∈ 𝑋 ∧ 𝑢 ∈ 𝑆)) → ((◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ^ko 𝑅))
139		ineq1 3807	. . . . . . . . . . . . 13 ⊢ (𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) = ((◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) ∩ (𝑅 Cn 𝑆)))
140	139	eleq1d 2686	. . . . . . . . . . . 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)))
141	138, 140	syl5ibrcom 237	. . . . . . . . . . 11 ⊢ (((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	rexlimdvva 3038	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ^ko 𝑅)))
143	142	imp 445	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ∃𝑘 ∈ 𝑋 ∃𝑢 ∈ 𝑆 𝑥 = (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢)) → (𝑥 ∩ (𝑅 Cn 𝑆)) ∈ (𝑆 ^ko 𝑅))
144	88, 143	sylan2b 492	. . . . . . . 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))
145	85, 144	jaodan 826	. . . . . . 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))
146	66, 145	sylan2b 492	. . . . . 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))
147		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ ({(∪ 𝑆 ↑𝑚 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))) = (𝑥 ∈ ({(∪ 𝑆 ↑𝑚 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆)))
148	146, 147	fmptd 6385	. . . . 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))
149		frn 6053	. . . . 5 ⊢ ((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) → ran (𝑥 ∈ ({(∪ 𝑆 ↑𝑚 𝑋)} ∪ ran (𝑘 ∈ 𝑋, 𝑢 ∈ 𝑆 ↦ (◡(𝑤 ∈ (∪ 𝑆 ↑𝑚 𝑋) ↦ (𝑤‘𝑘)) “ 𝑢))) ↦ (𝑥 ∩ (𝑅 Cn 𝑆))) ⊆ (𝑆 ^ko 𝑅))
150	148, 149	syl 17	. . . 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))
151	65, 150	eqsstrd 3639	. . 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))
152		tgfiss 20795	. . 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))
153	56, 151, 152	syl2anc 693	. 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))
154	55, 153	eqsstrd 3639	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽 ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ^ko 𝑅))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑_𝑚 cmap 7857  Xcixp 7908  Fincfn 7955  ficfi 8316   ↾_t crest 16081  topGenctg 16098  ∏_tcpt 16099  Topctop 20698  TopOnctopon 20715   Cn ccn 21028  Compccmp 21189   ^_ko cxko 21364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-pt 16105  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cmp 21190  df-xko 21366

This theorem is referenced by:  xkopt  21458  xkopjcn  21459