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Theorem cnntr 21813
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:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑥,𝑌

Proof of Theorem cnntr
StepHypRef Expression
1 cnf2 21787 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expia 1113 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌))
3 elpwi 4549 . . . . . . 7 (𝑥 ∈ 𝒫 𝑌𝑥𝑌)
43adantl 482 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥𝑌)
5 toponuni 21452 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
65ad2antlr 723 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑌 = 𝐾)
74, 6sseqtrd 4006 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 𝐾)
8 eqid 2821 . . . . . . 7 𝐾 = 𝐾
98cnntri 21809 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐾) → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)))
109expcom 414 . . . . 5 (𝑥 𝐾 → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
117, 10syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
1211ralrimdva 3189 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
132, 12jcad 513 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)))))
14 toponss 21465 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝐾) → 𝑥𝑌)
15 velpw 4545 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝑌𝑥𝑌)
1614, 15sylibr 235 . . . . . . . . 9 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝐾) → 𝑥 ∈ 𝒫 𝑌)
1716ex 413 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → (𝑥𝐾𝑥 ∈ 𝒫 𝑌))
1817ad2antlr 723 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥𝐾𝑥 ∈ 𝒫 𝑌))
1918imim1d 82 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ 𝒫 𝑌 → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))) → (𝑥𝐾 → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)))))
20 topontop 21451 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2120ad3antrrr 726 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → 𝐽 ∈ Top)
22 cnvimass 5943 . . . . . . . . . . 11 (𝐹𝑥) ⊆ dom 𝐹
23 fdm 6516 . . . . . . . . . . . . 13 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
2423ad2antlr 723 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → dom 𝐹 = 𝑋)
25 toponuni 21452 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2625ad3antrrr 726 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → 𝑋 = 𝐽)
2724, 26eqtrd 2856 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → dom 𝐹 = 𝐽)
2822, 27sseqtrid 4018 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝐽)
29 eqid 2821 . . . . . . . . . . 11 𝐽 = 𝐽
3029ntrss2 21595 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑥) ⊆ 𝐽) → ((int‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥))
3121, 28, 30syl2anc 584 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((int‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥))
32 eqss 3981 . . . . . . . . . 10 (((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥) ↔ (((int‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥) ∧ (𝐹𝑥) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
3332baib 536 . . . . . . . . 9 (((int‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥) → (((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹𝑥) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
3431, 33syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → (((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹𝑥) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
3529isopn3 21604 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝐹𝑥) ⊆ 𝐽) → ((𝐹𝑥) ∈ 𝐽 ↔ ((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥)))
3621, 28, 35syl2anc 584 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((𝐹𝑥) ∈ 𝐽 ↔ ((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥)))
37 topontop 21451 . . . . . . . . . . . 12 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
3837ad3antlr 727 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → 𝐾 ∈ Top)
39 isopn3i 21620 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝑥𝐾) → ((int‘𝐾)‘𝑥) = 𝑥)
4038, 39sylancom 588 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((int‘𝐾)‘𝑥) = 𝑥)
4140imaeq2d 5923 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → (𝐹 “ ((int‘𝐾)‘𝑥)) = (𝐹𝑥))
4241sseq1d 3997 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)) ↔ (𝐹𝑥) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
4334, 36, 423bitr4rd 313 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)) ↔ (𝐹𝑥) ∈ 𝐽))
4443pm5.74da 800 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥𝐾 → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))) ↔ (𝑥𝐾 → (𝐹𝑥) ∈ 𝐽)))
4519, 44sylibd 240 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ 𝒫 𝑌 → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))) → (𝑥𝐾 → (𝐹𝑥) ∈ 𝐽)))
4645ralimdv2 3176 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)) → ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽))
4746imdistanda 572 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))) → (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
48 iscn 21773 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
4947, 48sylibrd 260 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾)))
5013, 49impbid 213 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3138  wss 3935  𝒫 cpw 4537   cuni 4832  ccnv 5548  dom cdm 5549  cima 5552  wf 6345  cfv 6349  (class class class)co 7145  Topctop 21431  TopOnctopon 21448  intcnt 21555   Cn ccn 21762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8398  df-top 21432  df-topon 21449  df-ntr 21558  df-cn 21765
This theorem is referenced by: (None)
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