Theorem iscnp2 23563

Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Mario Carneiro, 21-Aug-2015.)

Hypotheses

Ref	Expression
iscn.1	⊢ 𝑋 = ∪ 𝐽
iscn.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
iscnp2	⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝑋,𝑦   𝑥,𝐹,𝑦   𝑥,𝑃,𝑦   𝑥,𝑌,𝑦

Proof of Theorem iscnp2

Dummy variables 𝑓 𝑔 𝑗 𝑘 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 4285	. . . . . . 7 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ¬ ((𝐽 CnP 𝐾)‘𝑃) = ∅)
2		df-ov 7424	. . . . . . . . . 10 ⊢ (𝐽 CnP 𝐾) = ( CnP ‘⟨𝐽, 𝐾⟩)
3		ndmfv 6918	. . . . . . . . . 10 ⊢ (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ( CnP ‘⟨𝐽, 𝐾⟩) = ∅)
4	2, 3	eqtrid 2807	. . . . . . . . 9 ⊢ (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → (𝐽 CnP 𝐾) = ∅)
5	4	fveq1d 6888	. . . . . . . 8 ⊢ (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ((𝐽 CnP 𝐾)‘𝑃) = (∅‘𝑃))
6		0fv 6927	. . . . . . . 8 ⊢ (∅‘𝑃) = ∅
7	5, 6	eqtrdi 2811	. . . . . . 7 ⊢ (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ((𝐽 CnP 𝐾)‘𝑃) = ∅)
8	1, 7	nsyl2 142	. . . . . 6 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ⟨𝐽, 𝐾⟩ ∈ dom CnP )
9		df-cnp 23552	. . . . . . 7 ⊢ CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
10		ovex 7454	. . . . . . . . . . 11 ⊢ (∪ 𝑘 ↑m ∪ 𝑗) ∈ V
11		ssrab2 4005	. . . . . . . . . . 11 ⊢ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))} ⊆ (∪ 𝑘 ↑m ∪ 𝑗)
12	10, 11	elpwi2 5296	. . . . . . . . . 10 ⊢ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))} ∈ 𝒫 (∪ 𝑘 ↑m ∪ 𝑗)
13	12	rgenw 3080	. . . . . . . . 9 ⊢ ∀𝑥 ∈ ∪ 𝑗{𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))} ∈ 𝒫 (∪ 𝑘 ↑m ∪ 𝑗)
14		eqid 2760	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) = (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})
15	14	fmpt 7111	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ∪ 𝑗{𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))} ∈ 𝒫 (∪ 𝑘 ↑m ∪ 𝑗) ↔ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}):∪ 𝑗⟶𝒫 (∪ 𝑘 ↑m ∪ 𝑗))
16	13, 15	mpbi 233	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}):∪ 𝑗⟶𝒫 (∪ 𝑘 ↑m ∪ 𝑗)
17		vuniex 7757	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ V
18	10	pwex 5341	. . . . . . . 8 ⊢ 𝒫 (∪ 𝑘 ↑m ∪ 𝑗) ∈ V
19		fex2 7949	. . . . . . . 8 ⊢ (((𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}):∪ 𝑗⟶𝒫 (∪ 𝑘 ↑m ∪ 𝑗) ∧ ∪ 𝑗 ∈ V ∧ 𝒫 (∪ 𝑘 ↑m ∪ 𝑗) ∈ V) → (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) ∈ V)
20	16, 17, 18, 19	mp3an 1490	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) ∈ V
21	9, 20	dmmpo 8083	. . . . . 6 ⊢ dom CnP = (Top × Top)
22	8, 21	eleqtrdi 2870	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ⟨𝐽, 𝐾⟩ ∈ (Top × Top))
23		opelxp 5686	. . . . 5 ⊢ (⟨𝐽, 𝐾⟩ ∈ (Top × Top) ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
24	22, 23	sylib 221	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
25	24	simpld 500	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
26	24	simprd 501	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
27		elfvdm 6920	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
28		iscn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
29	28	toptopon 23241	. . . . . . . 8 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
30		iscn.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐾
31	30	toptopon 23241	. . . . . . . 8 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
32		cnpfval 23558	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑥 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))}))
33	29, 31, 32	syl2anb 610	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 CnP 𝐾) = (𝑥 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))}))
34	24, 33	syl 18	. . . . . 6 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 CnP 𝐾) = (𝑥 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))}))
35	34	dmeqd 5886	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → dom (𝐽 CnP 𝐾) = dom (𝑥 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))}))
36		ovex 7454	. . . . . . . 8 ⊢ (𝑌 ↑m 𝑋) ∈ V
37	36	rabex 5299	. . . . . . 7 ⊢ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))} ∈ V
38	37	rgenw 3080	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑋 {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))} ∈ V
39		dmmptg 6243	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))} ∈ V → dom (𝑥 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))}) = 𝑋)
40	38, 39	ax-mp 5	. . . . 5 ⊢ dom (𝑥 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))}) = 𝑋
41	35, 40	eqtrdi 2811	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → dom (𝐽 CnP 𝐾) = 𝑋)
42	27, 41	eleqtrd 2862	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ 𝑋)
43	25, 26, 42	3jca 1146	. 2 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋))
44		biid 264	. . 3 ⊢ (𝑃 ∈ 𝑋 ↔ 𝑃 ∈ 𝑋)
45		iscnp 23561	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
46	29, 31, 44, 45	syl3anb 1179	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
47	43, 46	biadanii 834	1 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556  ⟨cop 4834  ∪ cuni 4866   ↦ cmpt 5185   × cxp 5648  dom cdm 5650   “ cima 5653  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ↑_m cmap 8851  Topctop 23217  TopOnctopon 23234   CnP ccnp 23549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-map 8853  df-top 23218  df-topon 23235  df-cnp 23552

This theorem is used by:  cnptop1  23566  cnptop2  23567  cnprcl  23569  cnpf  23571  cnpimaex  23580  cnpnei  23588  cnpco  23591  cnprest  23613  cnprest2  23614