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Theorem cndis 21906
 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 5917 . . . . . . . 8 (𝑓𝑥) ⊆ dom 𝑓
2 fdm 6496 . . . . . . . . 9 (𝑓:𝐴𝑋 → dom 𝑓 = 𝐴)
32adantl 485 . . . . . . . 8 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → dom 𝑓 = 𝐴)
41, 3sseqtrid 3967 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ⊆ 𝐴)
5 elpw2g 5212 . . . . . . . 8 (𝐴𝑉 → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
65ad2antrr 725 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
74, 6mpbird 260 . . . . . 6 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
87ralrimivw 3150 . . . . 5 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)
98ex 416 . . . 4 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴))
109pm4.71d 565 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
11 toponmax 21541 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
12 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
13 elmapg 8405 . . . 4 ((𝑋𝐽𝐴𝑉) → (𝑓 ∈ (𝑋m 𝐴) ↔ 𝑓:𝐴𝑋))
1411, 12, 13syl2anr 599 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋m 𝐴) ↔ 𝑓:𝐴𝑋))
15 distopon 21612 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
16 iscn 21850 . . . 4 ((𝒫 𝐴 ∈ (TopOn‘𝐴) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1715, 16sylan 583 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1810, 14, 173bitr4rd 315 . 2 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ 𝑓 ∈ (𝑋m 𝐴)))
1918eqrdv 2796 1 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋m 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  𝒫 cpw 4497  ◡ccnv 5519  dom cdm 5520   “ cima 5523  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136   ↑m cmap 8392  TopOnctopon 21525   Cn ccn 21839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-map 8394  df-top 21509  df-topon 21526  df-cn 21842 This theorem is referenced by:  xkopt  22270  distgp  22714  efmndtmd  22716  symgtgp  22721
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