Theorem cnntr 23160

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 23134	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
2	1	3expia 1121	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌))
3		elpwi 4558	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑌 → 𝑥 ⊆ 𝑌)
4	3	adantl 481	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ 𝑌)
5		toponuni 22799	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
6	5	ad2antlr 727	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑌 = ∪ 𝐾)
7	4, 6	sseqtrd 3972	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 ⊆ ∪ 𝐾)
8		eqid 2729	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	cnntri 23156	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)))
10	9	expcom 413	. . . . 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 3129	. . 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 512	. 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 22812	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑌)
15		velpw 4556	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝑌 ↔ 𝑥 ⊆ 𝑌)
16	14, 15	sylibr 234	. . . . . . . . 9 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝒫 𝑌)
17	16	ex 412	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘𝑌) → (𝑥 ∈ 𝐾 → 𝑥 ∈ 𝒫 𝑌))
18	17	ad2antlr 727	. . . . . . 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 22798	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21	20	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ Top)
22		cnvimass 6033	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
23		fdm 6661	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
24	23	ad2antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → dom 𝐹 = 𝑋)
25		toponuni 22799	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	25	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑋 = ∪ 𝐽)
27	24, 26	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → dom 𝐹 = ∪ 𝐽)
28	22, 27	sseqtrid 3978	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽)
29		eqid 2729	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
30	29	ntrss2 22942	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽) → ((int‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥))
31	21, 28, 30	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → ((int‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥))
32		eqss 3951	. . . . . . . . . 10 ⊢ (((int‘𝐽)‘(◡𝐹 “ 𝑥)) = (◡𝐹 “ 𝑥) ↔ (((int‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ 𝑥) ∧ (◡𝐹 “ 𝑥) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))
33	32	baib 535	. . . . . . . . 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 22951	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑥) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ((int‘𝐽)‘(◡𝐹 “ 𝑥)) = (◡𝐹 “ 𝑥)))
36	21, 28, 35	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ((int‘𝐽)‘(◡𝐹 “ 𝑥)) = (◡𝐹 “ 𝑥)))
37		topontop 22798	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
38	37	ad3antlr 731	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ Top)
39		isopn3i 22967	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ 𝑥 ∈ 𝐾) → ((int‘𝐾)‘𝑥) = 𝑥)
40	38, 39	sylancom 588	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → ((int‘𝐾)‘𝑥) = 𝑥)
41	40	imaeq2d 6011	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑥)) = (◡𝐹 “ 𝑥))
42	41	sseq1d 3967	. . . . . . . 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 312	. . . . . . 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 803	. . . . . 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 239	. . . . 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 3138	. . . 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 571	. . 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 23120	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
49	47, 48	sylibrd 259	. 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 212	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  ^◡ccnv 5618  dom cdm 5619   “ cima 5622  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Topctop 22778  TopOnctopon 22795  intcnt 22902   Cn ccn 23109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-map 8755  df-top 22779  df-topon 22796  df-ntr 22905  df-cn 23112

This theorem is referenced by: (None)