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

Theorem cndis 23299
Description: Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cndis ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋m 𝐴))

Proof of Theorem cndis
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 6100 . . . . . . . 8 (𝑓𝑥) ⊆ dom 𝑓
2 fdm 6745 . . . . . . . . 9 (𝑓:𝐴𝑋 → dom 𝑓 = 𝐴)
32adantl 481 . . . . . . . 8 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → dom 𝑓 = 𝐴)
41, 3sseqtrid 4026 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ⊆ 𝐴)
5 elpw2g 5333 . . . . . . . 8 (𝐴𝑉 → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
65ad2antrr 726 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
74, 6mpbird 257 . . . . . 6 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
87ralrimivw 3150 . . . . 5 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)
98ex 412 . . . 4 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴))
109pm4.71d 561 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
11 toponmax 22932 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
12 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
13 elmapg 8879 . . . 4 ((𝑋𝐽𝐴𝑉) → (𝑓 ∈ (𝑋m 𝐴) ↔ 𝑓:𝐴𝑋))
1411, 12, 13syl2anr 597 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋m 𝐴) ↔ 𝑓:𝐴𝑋))
15 distopon 23004 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
16 iscn 23243 . . . 4 ((𝒫 𝐴 ∈ (TopOn‘𝐴) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1715, 16sylan 580 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1810, 14, 173bitr4rd 312 . 2 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ 𝑓 ∈ (𝑋m 𝐴)))
1918eqrdv 2735 1 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wss 3951  𝒫 cpw 4600  ccnv 5684  dom cdm 5685  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  TopOnctopon 22916   Cn ccn 23232
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235
This theorem is referenced by:  xkopt  23663  distgp  24107  efmndtmd  24109  symgtgp  24114
  Copyright terms: Public domain W3C validator