MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssidcn Structured version   Visualization version   GIF version

Theorem ssidcn 23279
Description: The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
ssidcn ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾𝐽))

Proof of Theorem ssidcn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iscn 23259 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
2 f1oi 6887 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1of 6849 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋𝑋
54biantrur 530 . . 3 (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
61, 5bitr4di 289 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
7 cnvresid 6647 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
87imaeq1i 6077 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
9 elssuni 4942 . . . . . . . . 9 (𝑥𝐾𝑥 𝐾)
109adantl 481 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥 𝐾)
11 toponuni 22936 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = 𝐾)
1211ad2antlr 727 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑋 = 𝐾)
1310, 12sseqtrrd 4037 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥𝑋)
14 resiima 6096 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1513, 14syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
168, 15eqtrid 2787 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1716eleq1d 2824 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
1817ralbidva 3174 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ ∀𝑥𝐾 𝑥𝐽))
19 dfss3 3984 . . 3 (𝐾𝐽 ↔ ∀𝑥𝐾 𝑥𝐽)
2018, 19bitr4di 289 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝐾𝐽))
216, 20bitrd 279 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963   cuni 4912   I cid 5582  ccnv 5688  cres 5691  cima 5692  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  TopOnctopon 22932   Cn ccn 23248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-top 22916  df-topon 22933  df-cn 23251
This theorem is referenced by:  idcn  23281  sshauslem  23396
  Copyright terms: Public domain W3C validator