Theorem cnindis 21096

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 4197	. . . . . . 7 ⊢ (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2		topontop 20718	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3	2	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → 𝐽 ∈ Top)
4		0opn 20709	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ∅ ∈ 𝐽)
6		imaeq2 5462	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = (◡𝑓 “ ∅))
7		ima0 5481	. . . . . . . . . . 11 ⊢ (◡𝑓 “ ∅) = ∅
8	6, 7	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = ∅)
9	8	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((◡𝑓 “ 𝑥) ∈ 𝐽 ↔ ∅ ∈ 𝐽))
10	5, 9	syl5ibrcom 237	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (𝑥 = ∅ → (◡𝑓 “ 𝑥) ∈ 𝐽))
11		fimacnv 6347	. . . . . . . . . . 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 20730	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
14	13	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → 𝑋 ∈ 𝐽)
15	12, 14	eqeltrd 2701	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (◡𝑓 “ 𝐴) ∈ 𝐽)
16		imaeq2 5462	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (◡𝑓 “ 𝑥) = (◡𝑓 “ 𝐴))
17	16	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((◡𝑓 “ 𝑥) ∈ 𝐽 ↔ (◡𝑓 “ 𝐴) ∈ 𝐽))
18	15, 17	syl5ibrcom 237	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (𝑥 = 𝐴 → (◡𝑓 “ 𝑥) ∈ 𝐽))
19	10, 18	jaod 395	. . . . . . 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 2965	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)
22	21	ex 450	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓:𝑋⟶𝐴 → ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽))
23	22	pm4.71d 666	. . 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 7870	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐽) → (𝑓 ∈ (𝐴 ↑𝑚 𝑋) ↔ 𝑓:𝑋⟶𝐴))
26	24, 13, 25	syl2anr 495	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐴 ↑𝑚 𝑋) ↔ 𝑓:𝑋⟶𝐴))
27		indistopon 20805	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28		iscn 21039	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
29	27, 28	sylan2 491	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
30	23, 26, 29	3bitr4rd 301	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ 𝑓 ∈ (𝐴 ↑𝑚 𝑋)))
31	30	eqrdv 2620	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴 ↑𝑚 𝑋))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∅c0 3915  {cpr 4179  ^◡ccnv 5113   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Topctop 20698  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:  indishmph  21601  indistgp  21904  indispconn  31216