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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋 ↑𝑚 𝐴))

Proof of Theorem cndis

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 5635	. . . . . . . 8 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓
2		fdm 6204	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝑋 → dom 𝑓 = 𝐴)
3	2	adantl 473	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → dom 𝑓 = 𝐴)
4	1, 3	syl5sseq 3786	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → (◡𝑓 “ 𝑥) ⊆ 𝐴)
5		elpw2g 4968	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((◡𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑥) ⊆ 𝐴))
6	5	ad2antrr 764	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → ((◡𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑥) ⊆ 𝐴))
7	4, 6	mpbird 247	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)
8	7	ralrimivw 3097	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)
9	8	ex 449	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴))
10	9	pm4.71d 669	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 ↔ (𝑓:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)))
11		toponmax 20924	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
12		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
13		elmapg 8028	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝑋 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝑋))
14	11, 12, 13	syl2anr 496	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝑋))
15		distopon 20995	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
16		iscn 21233	. . . 4 ⊢ ((𝒫 𝐴 ∈ (TopOn‘𝐴) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)))
17	15, 16	sylan 489	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)))
18	10, 14, 17	3bitr4rd 301	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ 𝑓 ∈ (𝑋 ↑𝑚 𝐴)))
19	18	eqrdv 2750	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋 ↑𝑚 𝐴))

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