Theorem cnindis 23276

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 4580	. . . . . . 7 ⊢ (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2		topontop 22897	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3	2	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → 𝐽 ∈ Top)
4		0opn 22888	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ∅ ∈ 𝐽)
6		imaeq2 6009	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = (◡𝑓 “ ∅))
7		ima0 6030	. . . . . . . . . . 11 ⊢ (◡𝑓 “ ∅) = ∅
8	6, 7	eqtrdi 2790	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = ∅)
9	8	eleq1d 2824	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((◡𝑓 “ 𝑥) ∈ 𝐽 ↔ ∅ ∈ 𝐽))
10	5, 9	syl5ibrcom 248	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (𝑥 = ∅ → (◡𝑓 “ 𝑥) ∈ 𝐽))
11		fimacnv 6678	. . . . . . . . . . 11 ⊢ (𝑓:𝑋⟶𝐴 → (◡𝑓 “ 𝐴) = 𝑋)
12	11	adantl 482	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (◡𝑓 “ 𝐴) = 𝑋)
13		toponmax 22910	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
14	13	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → 𝑋 ∈ 𝐽)
15	12, 14	eqeltrd 2839	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (◡𝑓 “ 𝐴) ∈ 𝐽)
16		imaeq2 6009	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (◡𝑓 “ 𝑥) = (◡𝑓 “ 𝐴))
17	16	eleq1d 2824	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((◡𝑓 “ 𝑥) ∈ 𝐽 ↔ (◡𝑓 “ 𝐴) ∈ 𝐽))
18	15, 17	syl5ibrcom 248	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (𝑥 = 𝐴 → (◡𝑓 “ 𝑥) ∈ 𝐽))
19	10, 18	jaod 865	. . . . . . 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 3130	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)
22	21	ex 413	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓:𝑋⟶𝐴 → ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽))
23	22	pm4.71d 566	. . 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 8777	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐽) → (𝑓 ∈ (𝐴 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝐴))
26	24, 13, 25	syl2anr 603	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐴 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝐴))
27		indistopon 22985	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28		iscn 23219	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
29	27, 28	sylan2 599	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
30	23, 26, 29	3bitr4rd 313	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ 𝑓 ∈ (𝐴 ↑m 𝑋)))
31	30	eqrdv 2737	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴 ↑m 𝑋))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∅c0 4262  {cpr 4558  ^◡ccnv 5618   “ cima 5622  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357   ↑_m cmap 8764  Topctop 22877  TopOnctopon 22894   Cn ccn 23208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8766  df-top 22878  df-topon 22895  df-cn 23211

This theorem is referenced by:  indishmph  23782  indistgp  24084  indispconn  35471