Theorem cndis 14964

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 5099	. . . . . . . 8 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓
2		fdm 5488	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝑋 → dom 𝑓 = 𝐴)
3	2	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → dom 𝑓 = 𝐴)
4	1, 3	sseqtrid 3277	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → (◡𝑓 “ 𝑥) ⊆ 𝐴)
5		elpw2g 4246	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((◡𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑥) ⊆ 𝐴))
6	5	ad2antrr 488	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → ((◡𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑥) ⊆ 𝐴))
7	4, 6	mpbird 167	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)
8	7	ralrimivw 2606	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)
9	8	ex 115	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴))
10	9	pm4.71d 393	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 ↔ (𝑓:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)))
11		toponmax 14748	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
12		id 19	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
13		elmapg 6829	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝑋 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝑋))
14	11, 12, 13	syl2anr 290	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝑋))
15		distopon 14810	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
16		iscn 14920	. . . 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 283	. . 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 221	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ 𝑓 ∈ (𝑋 ↑𝑚 𝐴)))
19	18	eqrdv 2229	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋 ↑𝑚 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ⊆ wss 3200  𝒫 cpw 3652  ^◡ccnv 4724  dom cdm 4725   “ cima 4728  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017   ↑_𝑚 cmap 6816  TopOnctopon 14733   Cn ccn 14908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-top 14721  df-topon 14734  df-cn 14911

This theorem is referenced by: (None)