Theorem cnindis 23186

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 4616	. . . . . . 7 ⊢ (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2		topontop 22807	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3	2	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → 𝐽 ∈ Top)
4		0opn 22798	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ∅ ∈ 𝐽)
6		imaeq2 6030	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = (◡𝑓 “ ∅))
7		ima0 6051	. . . . . . . . . . 11 ⊢ (◡𝑓 “ ∅) = ∅
8	6, 7	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = ∅)
9	8	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((◡𝑓 “ 𝑥) ∈ 𝐽 ↔ ∅ ∈ 𝐽))
10	5, 9	syl5ibrcom 247	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (𝑥 = ∅ → (◡𝑓 “ 𝑥) ∈ 𝐽))
11		fimacnv 6713	. . . . . . . . . . 11 ⊢ (𝑓:𝑋⟶𝐴 → (◡𝑓 “ 𝐴) = 𝑋)
12	11	adantl 481	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (◡𝑓 “ 𝐴) = 𝑋)
13		toponmax 22820	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
14	13	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → 𝑋 ∈ 𝐽)
15	12, 14	eqeltrd 2829	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (◡𝑓 “ 𝐴) ∈ 𝐽)
16		imaeq2 6030	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (◡𝑓 “ 𝑥) = (◡𝑓 “ 𝐴))
17	16	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((◡𝑓 “ 𝑥) ∈ 𝐽 ↔ (◡𝑓 “ 𝐴) ∈ 𝐽))
18	15, 17	syl5ibrcom 247	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (𝑥 = 𝐴 → (◡𝑓 “ 𝑥) ∈ 𝐽))
19	10, 18	jaod 859	. . . . . . 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 3125	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)
22	21	ex 412	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓:𝑋⟶𝐴 → ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽))
23	22	pm4.71d 561	. . 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 8815	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐽) → (𝑓 ∈ (𝐴 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝐴))
26	24, 13, 25	syl2anr 597	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐴 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝐴))
27		indistopon 22895	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28		iscn 23129	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
29	27, 28	sylan2 593	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
30	23, 26, 29	3bitr4rd 312	. 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 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∅c0 4299  {cpr 4594  ^◡ccnv 5640   “ cima 5644  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  Topctop 22787  TopOnctopon 22804   Cn ccn 23118

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-top 22788  df-topon 22805  df-cn 23121

This theorem is referenced by:  indishmph  23692  indistgp  23994  indispconn  35228