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Theorem cnntr 12433
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 12413 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expia 1184 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌))
3 elpwi 3524 . . . . . . 7 (𝑥 ∈ 𝒫 𝑌𝑥𝑌)
43adantl 275 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥𝑌)
5 toponuni 12221 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
65ad2antlr 481 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑌 = 𝐾)
74, 6sseqtrd 3140 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → 𝑥 𝐾)
8 eqid 2140 . . . . . . 7 𝐾 = 𝐾
98cnntri 12432 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐾) → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)))
109expcom 115 . . . . 5 (𝑥 𝐾 → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
117, 10syl 14 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝒫 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
1211ralrimdva 2515 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
132, 12jcad 305 . 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 12232 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝐾) → 𝑥𝑌)
15 velpw 3522 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝑌𝑥𝑌)
1614, 15sylibr 133 . . . . . . . . 9 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝐾) → 𝑥 ∈ 𝒫 𝑌)
1716ex 114 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → (𝑥𝐾𝑥 ∈ 𝒫 𝑌))
1817ad2antlr 481 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥𝐾𝑥 ∈ 𝒫 𝑌))
1918imim1d 75 . . . . . 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 12220 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2120ad3antrrr 484 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → 𝐽 ∈ Top)
22 cnvimass 4910 . . . . . . . . . . 11 (𝐹𝑥) ⊆ dom 𝐹
23 fdm 5286 . . . . . . . . . . . . 13 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
2423ad2antlr 481 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → dom 𝐹 = 𝑋)
25 toponuni 12221 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2625ad3antrrr 484 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → 𝑋 = 𝐽)
2724, 26eqtrd 2173 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → dom 𝐹 = 𝐽)
2822, 27sseqtrid 3152 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝐽)
29 eqid 2140 . . . . . . . . . . 11 𝐽 = 𝐽
3029ntrss2 12329 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑥) ⊆ 𝐽) → ((int‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥))
3121, 28, 30syl2anc 409 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((int‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥))
32 eqss 3117 . . . . . . . . . 10 (((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥) ↔ (((int‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥) ∧ (𝐹𝑥) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
3332baib 905 . . . . . . . . 9 (((int‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹𝑥) → (((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹𝑥) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
3431, 33syl 14 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → (((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥) ↔ (𝐹𝑥) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
3529isopn3 12333 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝐹𝑥) ⊆ 𝐽) → ((𝐹𝑥) ∈ 𝐽 ↔ ((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥)))
3621, 28, 35syl2anc 409 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((𝐹𝑥) ∈ 𝐽 ↔ ((int‘𝐽)‘(𝐹𝑥)) = (𝐹𝑥)))
37 topontop 12220 . . . . . . . . . . . 12 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
3837ad3antlr 485 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → 𝐾 ∈ Top)
39 isopn3i 12343 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝑥𝐾) → ((int‘𝐾)‘𝑥) = 𝑥)
4038, 39sylancom 417 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((int‘𝐾)‘𝑥) = 𝑥)
4140imaeq2d 4889 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → (𝐹 “ ((int‘𝐾)‘𝑥)) = (𝐹𝑥))
4241sseq1d 3131 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)) ↔ (𝐹𝑥) ⊆ ((int‘𝐽)‘(𝐹𝑥))))
4334, 36, 423bitr4rd 220 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝐾) → ((𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)) ↔ (𝐹𝑥) ∈ 𝐽))
4443pm5.74da 440 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥𝐾 → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))) ↔ (𝑥𝐾 → (𝐹𝑥) ∈ 𝐽)))
4519, 44sylibd 148 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ 𝒫 𝑌 → (𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥))) → (𝑥𝐾 → (𝐹𝑥) ∈ 𝐽)))
4645ralimdv2 2505 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)) → ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽))
4746imdistanda 445 . . 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 12405 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
4947, 48sylibrd 168 . 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 128 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 set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  wral 2417  wss 3076  𝒫 cpw 3515   cuni 3744  ccnv 4546  dom cdm 4547  cima 4550  wf 5127  cfv 5131  (class class class)co 5782  Topctop 12203  TopOnctopon 12216  intcnt 12301   Cn ccn 12393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-top 12204  df-topon 12217  df-ntr 12304  df-cn 12396
This theorem is referenced by: (None)
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