Theorem cndis 22188

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 5949	. . . . . . . 8 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓
2		fdm 6554	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝑋 → dom 𝑓 = 𝐴)
3	2	adantl 485	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → dom 𝑓 = 𝐴)
4	1, 3	sseqtrid 3953	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → (◡𝑓 “ 𝑥) ⊆ 𝐴)
5		elpw2g 5237	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((◡𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑥) ⊆ 𝐴))
6	5	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → ((◡𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑥) ⊆ 𝐴))
7	4, 6	mpbird 260	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)
8	7	ralrimivw 3106	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)
9	8	ex 416	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴))
10	9	pm4.71d 565	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 ↔ (𝑓:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)))
11		toponmax 21823	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
12		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
13		elmapg 8521	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝑋 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝑋))
14	11, 12, 13	syl2anr 600	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝑋))
15		distopon 21894	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
16		iscn 22132	. . . 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 583	. . 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 315	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ 𝑓 ∈ (𝑋 ↑m 𝐴)))
19	18	eqrdv 2735	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋 ↑m 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061   ⊆ wss 3866  𝒫 cpw 4513  ^◡ccnv 5550  dom cdm 5551   “ cima 5554  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ↑_m cmap 8508  TopOnctopon 21807   Cn ccn 22121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-top 21791  df-topon 21808  df-cn 22124

This theorem is referenced by:  xkopt  22552  distgp  22996  efmndtmd  22998  symgtgp  23003