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Theorem cndis 21289
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 5635 . . . . . . . 8 (𝑓𝑥) ⊆ dom 𝑓
2 fdm 6204 . . . . . . . . 9 (𝑓:𝐴𝑋 → dom 𝑓 = 𝐴)
32adantl 473 . . . . . . . 8 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → dom 𝑓 = 𝐴)
41, 3syl5sseq 3786 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ⊆ 𝐴)
5 elpw2g 4968 . . . . . . . 8 (𝐴𝑉 → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
65ad2antrr 764 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
74, 6mpbird 247 . . . . . 6 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
87ralrimivw 3097 . . . . 5 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)
98ex 449 . . . 4 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴))
109pm4.71d 669 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
11 toponmax 20924 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
12 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
13 elmapg 8028 . . . 4 ((𝑋𝐽𝐴𝑉) → (𝑓 ∈ (𝑋𝑚 𝐴) ↔ 𝑓:𝐴𝑋))
1411, 12, 13syl2anr 496 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋𝑚 𝐴) ↔ 𝑓:𝐴𝑋))
15 distopon 20995 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
16 iscn 21233 . . . 4 ((𝒫 𝐴 ∈ (TopOn‘𝐴) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1715, 16sylan 489 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1810, 14, 173bitr4rd 301 . 2 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ 𝑓 ∈ (𝑋𝑚 𝐴)))
1918eqrdv 2750 1 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  wss 3707  𝒫 cpw 4294  ccnv 5257  dom cdm 5258  cima 5261  wf 6037  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  TopOnctopon 20909   Cn ccn 21222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-map 8017  df-top 20893  df-topon 20910  df-cn 21225
This theorem is referenced by:  xkopt  21652  distgp  22096  symgtgp  22098
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