Theorem cnindis 21572

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 4488	. . . . . . 7 ⊢ (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2		topontop 21193	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3	2	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → 𝐽 ∈ Top)
4		0opn 21184	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → ∅ ∈ 𝐽)
6		imaeq2 5794	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = (◡𝑓 “ ∅))
7		ima0 5813	. . . . . . . . . . 11 ⊢ (◡𝑓 “ ∅) = ∅
8	6, 7	syl6eq 2845	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (◡𝑓 “ 𝑥) = ∅)
9	8	eleq1d 2865	. . . . . . . . 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 6695	. . . . . . . . . . 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 21206	. . . . . . . . . . 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 2881	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝑋⟶𝐴) → (◡𝑓 “ 𝐴) ∈ 𝐽)
16		imaeq2 5794	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (◡𝑓 “ 𝑥) = (◡𝑓 “ 𝐴))
17	16	eleq1d 2865	. . . . . . . . 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 854	. . . . . . 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 3146	. . . . 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 562	. . 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 8260	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐽) → (𝑓 ∈ (𝐴 ↑𝑚 𝑋) ↔ 𝑓:𝑋⟶𝐴))
26	24, 13, 25	syl2anr 596	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐴 ↑𝑚 𝑋) ↔ 𝑓:𝑋⟶𝐴))
27		indistopon 21281	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28		iscn 21515	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋⟶𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (◡𝑓 “ 𝑥) ∈ 𝐽)))
29	27, 28	sylan2 592	. . 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 2791	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴 ↑𝑚 𝑋))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   = wceq 1520   ∈ wcel 2079  ∀wral 3103  ∅c0 4206  {cpr 4468  ^◡ccnv 5434   “ cima 5438  ⟶wf 6213  ‘cfv 6217  (class class class)co 7007   ↑_𝑚 cmap 8247  Topctop 21173  TopOnctopon 21190   Cn ccn 21504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-map 8249  df-top 21174  df-topon 21191  df-cn 21507

This theorem is referenced by:  indishmph  22078  indistgp  22380  indispconn  32045