Theorem cnntr 23337

Description: Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
cnntr	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)))))

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

Proof of Theorem cnntr

Step	Hyp	Ref	Expression
1		cnf2 23311	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
2	1	3expia 1135	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌))
3		elpwi 4564	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑌 → 𝑥 ⊆ 𝑌)
4	3	adantl 485	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ 𝑌)
5		toponuni 22976	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
6	5	ad2antlr 737	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑌 = ∪ 𝐾)
7	4, 6	sseqtrd 3974	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ ∪ 𝐾)
8		eqid 2764	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	cnntri 23333	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)))
10	9	expcom 417	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐾 → (𝐹 ∈ (𝐽 Cn 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))
11	7, 10	syl 17	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))
12	11	ralrimdva 3164	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))
13	2, 12	jcad 520	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)))))
14		toponss 22989	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑌)
15		velpw 4562	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝑌 ↔ 𝑥 ⊆ 𝑌)
16	14, 15	sylibr 236	. . . . . . . . 9 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝒫 𝑌)
17	16	ex 416	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘𝑌) → (𝑥 ∈ 𝐾 → 𝑥 ∈ 𝒫 𝑌))
18	17	ad2antlr 737	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ 𝐾 → 𝑥 ∈ 𝒫 𝑌))
19	18	imim1d 82	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ 𝒫 𝑌 → (◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))) → (𝑥 ∈ 𝐾 → (◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)))))
20		topontop 22975	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21	20	ad3antrrr 740	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ Top)
22		cnvimass 6073	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
23		fdm 6703	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
24	23	ad2antlr 737	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → dom 𝐹 = 𝑋)
25		toponuni 22976	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	25	ad3antrrr 740	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑋 = ∪ 𝐽)
27	24, 26	eqtrd 2799	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → dom 𝐹 = ∪ 𝐽)
28	22, 27	sseqtrid 3980	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽)
29		eqid 2764	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
30	29	ntrss2 23119	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽) → ((int‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥))
31	21, 28, 30	syl2anc 593	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → ((int‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥))
32		eqss 3953	. . . . . . . . . 10 ⊢ (((int‘𝐽)‘(◡𝐹 “ 𝑥)) = (◡𝐹 “ 𝑥) ↔ (((int‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥) ∧ (◡𝐹 “ 𝑥) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))
33	32	baib 543	. . . . . . . . 9 ⊢ (((int‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥) → (((int‘𝐽)‘(◡𝐹 “ 𝑥)) = (◡𝐹 “ 𝑥) ↔ (◡𝐹 “ 𝑥) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))
34	31, 33	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → (((int‘𝐽)‘(◡𝐹 “ 𝑥)) = (◡𝐹 “ 𝑥) ↔ (◡𝐹 “ 𝑥) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))
35	29	isopn3 23128	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ((int‘𝐽)‘(◡𝐹 “ 𝑥)) = (◡𝐹 “ 𝑥)))
36	21, 28, 35	syl2anc 593	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ((int‘𝐽)‘(◡𝐹 “ 𝑥)) = (◡𝐹 “ 𝑥)))
37		topontop 22975	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
38	37	ad3antlr 741	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ Top)
39		isopn3i 23144	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ 𝑥 ∈ 𝐾) → ((int‘𝐾)‘𝑥) = 𝑥)
40	38, 39	sylancom 597	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → ((int‘𝐾)‘𝑥) = 𝑥)
41	40	imaeq2d 6051	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑥)) = (◡𝐹 “ 𝑥))
42	41	sseq1d 3969	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → ((◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)) ↔ (◡𝐹 “ 𝑥) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))
43	34, 36, 42	3bitr4rd 314	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → ((◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)) ↔ (◡𝐹 “ 𝑥) ∈ 𝐽))
44	43	pm5.74da 813	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ 𝐾 → (◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))) ↔ (𝑥 ∈ 𝐾 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
45	19, 44	sylibd 241	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ 𝒫 𝑌 → (◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))) → (𝑥 ∈ 𝐾 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
46	45	ralimdv2 3173	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)) → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
47	46	imdistanda 579	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
48		iscn 23297	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
49	47, 48	sylibrd 261	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾)))
50	13, 49	impbid 214	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ⊆ wss 3906  𝒫 cpw 4557  ∪ cuni 4867  ^◡ccnv 5648  dom cdm 5649   “ cima 5652  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  Topctop 22955  TopOnctopon 22972  intcnt 23079   Cn ccn 23286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-map 8812  df-top 22956  df-topon 22973  df-ntr 23082  df-cn 23289

This theorem is referenced by: (None)