Theorem cnindis 23016

Description: Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cnindis	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴 ↑m 𝑋))

Proof of Theorem cnindis

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

Step	Hyp	Ref	Expression
1		elpri 4649	. . . . . . 7 ⊢ (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2		topontop 22635	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3	2	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → 𝐽 ∈ Top)
4		0opn 22626	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ∅ ∈ 𝐽)
6		imaeq2 6054	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = (◡𝑓 “ ∅))
7		ima0 6075	. . . . . . . . . . 11 ⊢ (◡𝑓 “ ∅) = ∅
8	6, 7	eqtrdi 2786	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = ∅)
9	8	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((◡𝑓 “ 𝑥) ∈ 𝐽 ↔ ∅ ∈ 𝐽))
10	5, 9	syl5ibrcom 246	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (𝑥 = ∅ → (◡𝑓 “ 𝑥) ∈ 𝐽))
11		fimacnv 6738	. . . . . . . . . . 11 ⊢ (𝑓:𝑋⟶𝐴 → (◡𝑓 “ 𝐴) = 𝑋)
12	11	adantl 480	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (◡𝑓 “ 𝐴) = 𝑋)
13		toponmax 22648	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
14	13	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → 𝑋 ∈ 𝐽)
15	12, 14	eqeltrd 2831	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (◡𝑓 “ 𝐴) ∈ 𝐽)
16		imaeq2 6054	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (◡𝑓 “ 𝑥) = (◡𝑓 “ 𝐴))
17	16	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((◡𝑓 “ 𝑥) ∈ 𝐽 ↔ (◡𝑓 “ 𝐴) ∈ 𝐽))
18	15, 17	syl5ibrcom 246	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (𝑥 = 𝐴 → (◡𝑓 “ 𝑥) ∈ 𝐽))
19	10, 18	jaod 855	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → (◡𝑓 “ 𝑥) ∈ 𝐽))
20	1, 19	syl5 34	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (𝑥 ∈ {∅, 𝐴} → (◡𝑓 “ 𝑥) ∈ 𝐽))
21	20	ralrimiv 3143	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)
22	21	ex 411	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓:𝑋⟶𝐴 → ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽))
23	22	pm4.71d 560	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓:𝑋⟶𝐴 ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
24		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
25		elmapg 8835	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐽) → (𝑓 ∈ (𝐴 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝐴))
26	24, 13, 25	syl2anr 595	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐴 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝐴))
27		indistopon 22724	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28		iscn 22959	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
29	27, 28	sylan2 591	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
30	23, 26, 29	3bitr4rd 311	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ 𝑓 ∈ (𝐴 ↑m 𝑋)))
31	30	eqrdv 2728	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴 ↑m 𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∅c0 4321  {cpr 4629  ^◡ccnv 5674   “ cima 5678  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411   ↑_m cmap 8822  Topctop 22615  TopOnctopon 22632   Cn ccn 22948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-top 22616  df-topon 22633  df-cn 22951

This theorem is referenced by:  indishmph  23522  indistgp  23824  indispconn  34523