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Theorem cndis 23286
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 6091 . . . . . . . 8 (𝑓𝑥) ⊆ dom 𝑓
2 fdm 6737 . . . . . . . . 9 (𝑓:𝐴𝑋 → dom 𝑓 = 𝐴)
32adantl 480 . . . . . . . 8 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → dom 𝑓 = 𝐴)
41, 3sseqtrid 4032 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ⊆ 𝐴)
5 elpw2g 5351 . . . . . . . 8 (𝐴𝑉 → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
65ad2antrr 724 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
74, 6mpbird 256 . . . . . 6 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
87ralrimivw 3140 . . . . 5 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)
98ex 411 . . . 4 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴))
109pm4.71d 560 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
11 toponmax 22919 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
12 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
13 elmapg 8868 . . . 4 ((𝑋𝐽𝐴𝑉) → (𝑓 ∈ (𝑋m 𝐴) ↔ 𝑓:𝐴𝑋))
1411, 12, 13syl2anr 595 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋m 𝐴) ↔ 𝑓:𝐴𝑋))
15 distopon 22991 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
16 iscn 23230 . . . 4 ((𝒫 𝐴 ∈ (TopOn‘𝐴) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1715, 16sylan 578 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1810, 14, 173bitr4rd 311 . 2 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ 𝑓 ∈ (𝑋m 𝐴)))
1918eqrdv 2724 1 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  wss 3947  𝒫 cpw 4607  ccnv 5681  dom cdm 5682  cima 5685  wf 6550  cfv 6554  (class class class)co 7424  m cmap 8855  TopOnctopon 22903   Cn ccn 23219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-top 22887  df-topon 22904  df-cn 23222
This theorem is referenced by:  xkopt  23650  distgp  24094  efmndtmd  24096  symgtgp  24101
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