Theorem iscnp2 23149

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 7344	. . . . . . . . . 10 ⊢ (𝐽 CnP 𝐾) = ( CnP ‘⟨𝐽, 𝐾⟩)
3		ndmfv 6849	. . . . . . . . . 10 ⊢ (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ( CnP ‘⟨𝐽, 𝐾⟩) = ∅)
4	2, 3	eqtrid 2778	. . . . . . . . 9 ⊢ (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → (𝐽 CnP 𝐾) = ∅)
5	4	fveq1d 6819	. . . . . . . 8 ⊢ (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ((𝐽 CnP 𝐾)‘𝑃) = (∅‘𝑃))
6		0fv 6858	. . . . . . . 8 ⊢ (∅‘𝑃) = ∅
7	5, 6	eqtrdi 2782	. . . . . . 7 ⊢ (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ((𝐽 CnP 𝐾)‘𝑃) = ∅)
8	1, 7	nsyl2 141	. . . . . 6 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ⟨𝐽, 𝐾⟩ ∈ dom CnP )
9		df-cnp 23138	. . . . . . 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 7374	. . . . . . . . . . 11 ⊢ (∪ 𝑘 ↑m ∪ 𝑗) ∈ V
11		ssrab2 4025	. . . . . . . . . . 11 ⊢ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))} ⊆ (∪ 𝑘 ↑m ∪ 𝑗)
12	10, 11	elpwi2 5268	. . . . . . . . . 10 ⊢ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))} ∈ 𝒫 (∪ 𝑘 ↑m ∪ 𝑗)
13	12	rgenw 3051	. . . . . . . . 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 2731	. . . . . . . . . 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 7038	. . . . . . . . 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 230	. . . . . . . 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 7667	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ V
18	10	pwex 5313	. . . . . . . 8 ⊢ 𝒫 (∪ 𝑘 ↑m ∪ 𝑗) ∈ V
19		fex2 7861	. . . . . . . 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 1463	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}) ∈ V
21	9, 20	dmmpo 7998	. . . . . 6 ⊢ dom CnP = (Top × Top)
22	8, 21	eleqtrdi 2841	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ⟨𝐽, 𝐾⟩ ∈ (Top × Top))
23		opelxp 5647	. . . . 5 ⊢ (⟨𝐽, 𝐾⟩ ∈ (Top × Top) ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
24	22, 23	sylib 218	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
25	24	simpld 494	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
26	24	simprd 495	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
27		elfvdm 6851	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
28		iscn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
29	28	toptopon 22827	. . . . . . . 8 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
30		iscn.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐾
31	30	toptopon 22827	. . . . . . . 8 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
32		cnpfval 23144	. . . . . . . 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 598	. . . . . . 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 17	. . . . . 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 5840	. . . . 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 7374	. . . . . . . 8 ⊢ (𝑌 ↑m 𝑋) ∈ V
37	36	rabex 5272	. . . . . . 7 ⊢ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))} ∈ V
38	37	rgenw 3051	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑋 {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))} ∈ V
39		dmmptg 6184	. . . . . 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 2782	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → dom (𝐽 CnP 𝐾) = 𝑋)
42	27, 41	eleqtrd 2833	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ 𝑋)
43	25, 26, 42	3jca 1128	. 2 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋))
44		biid 261	. . 3 ⊢ (𝑃 ∈ 𝑋 ↔ 𝑃 ∈ 𝑋)
45		iscnp 23147	. . 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 1161	. 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 821	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)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ⊆ wss 3897  ∅c0 4278  𝒫 cpw 4545  ⟨cop 4577  ∪ cuni 4854   ↦ cmpt 5167   × cxp 5609  dom cdm 5611   “ cima 5614  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341   ↑_m cmap 8745  Topctop 22803  TopOnctopon 22820   CnP ccnp 23135

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747  df-top 22804  df-topon 22821  df-cnp 23138

This theorem is referenced by:  cnptop1  23152  cnptop2  23153  cnprcl  23155  cnpf  23157  cnpimaex  23166  cnpnei  23174  cnpco  23177  cnprest  23199  cnprest2  23200