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Theorem cndis 21306
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 5695 . . . . . . . 8 (𝑓𝑥) ⊆ dom 𝑓
2 fdm 6260 . . . . . . . . 9 (𝑓:𝐴𝑋 → dom 𝑓 = 𝐴)
32adantl 469 . . . . . . . 8 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → dom 𝑓 = 𝐴)
41, 3syl5sseq 3850 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ⊆ 𝐴)
5 elpw2g 5019 . . . . . . . 8 (𝐴𝑉 → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
65ad2antrr 708 . . . . . . 7 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴))
74, 6mpbird 248 . . . . . 6 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
87ralrimivw 3155 . . . . 5 (((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴𝑋) → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)
98ex 399 . . . 4 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 → ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴))
109pm4.71d 553 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴𝑋 ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
11 toponmax 20941 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
12 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
13 elmapg 8101 . . . 4 ((𝑋𝐽𝐴𝑉) → (𝑓 ∈ (𝑋𝑚 𝐴) ↔ 𝑓:𝐴𝑋))
1411, 12, 13syl2anr 586 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋𝑚 𝐴) ↔ 𝑓:𝐴𝑋))
15 distopon 21012 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
16 iscn 21250 . . . 4 ((𝒫 𝐴 ∈ (TopOn‘𝐴) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1715, 16sylan 571 . . 3 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴𝑋 ∧ ∀𝑥𝐽 (𝑓𝑥) ∈ 𝒫 𝐴)))
1810, 14, 173bitr4rd 303 . 2 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ 𝑓 ∈ (𝑋𝑚 𝐴)))
1918eqrdv 2804 1 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  wss 3769  𝒫 cpw 4351  ccnv 5310  dom cdm 5311  cima 5314  wf 6093  cfv 6097  (class class class)co 6870  𝑚 cmap 8088  TopOnctopon 20925   Cn ccn 21239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-map 8090  df-top 20909  df-topon 20926  df-cn 21242
This theorem is referenced by:  xkopt  21669  distgp  22113  symgtgp  22115
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