Theorem cndis 21095

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 5485	. . . . . . . 8 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓
2		fdm 6051	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝑋 → dom 𝑓 = 𝐴)
3	2	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → dom 𝑓 = 𝐴)
4	1, 3	syl5sseq 3653	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → (◡𝑓 “ 𝑥) ⊆ 𝐴)
5		elpw2g 4827	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((◡𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑥) ⊆ 𝐴))
6	5	ad2antrr 762	. . . . . . 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 2967	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)
9	8	ex 450	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴))
10	9	pm4.71d 666	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 ↔ (𝑓:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)))
11		toponmax 20730	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
12		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
13		elmapg 7870	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝑋 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝑋))
14	11, 12, 13	syl2anr 495	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝑋))
15		distopon 20801	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
16		iscn 21039	. . . 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 488	. . 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 2620	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋 ↑𝑚 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  𝒫 cpw 4158  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031

This theorem is referenced by:  xkopt  21458  distgp  21903  symgtgp  21905