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Theorem cnindis 23300
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) ∧ 𝐴𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴m 𝑋))

Proof of Theorem cnindis
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpri 4649 . . . . . . 7 (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2 topontop 22919 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32ad2antrr 726 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝐽 ∈ Top)
4 0opn 22910 . . . . . . . . . 10 (𝐽 ∈ Top → ∅ ∈ 𝐽)
53, 4syl 17 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∅ ∈ 𝐽)
6 imaeq2 6074 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑓𝑥) = (𝑓 “ ∅))
7 ima0 6095 . . . . . . . . . . 11 (𝑓 “ ∅) = ∅
86, 7eqtrdi 2793 . . . . . . . . . 10 (𝑥 = ∅ → (𝑓𝑥) = ∅)
98eleq1d 2826 . . . . . . . . 9 (𝑥 = ∅ → ((𝑓𝑥) ∈ 𝐽 ↔ ∅ ∈ 𝐽))
105, 9syl5ibrcom 247 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = ∅ → (𝑓𝑥) ∈ 𝐽))
11 fimacnv 6758 . . . . . . . . . . 11 (𝑓:𝑋𝐴 → (𝑓𝐴) = 𝑋)
1211adantl 481 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) = 𝑋)
13 toponmax 22932 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1413ad2antrr 726 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝑋𝐽)
1512, 14eqeltrd 2841 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) ∈ 𝐽)
16 imaeq2 6074 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑓𝑥) = (𝑓𝐴))
1716eleq1d 2826 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑓𝑥) ∈ 𝐽 ↔ (𝑓𝐴) ∈ 𝐽))
1815, 17syl5ibrcom 247 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = 𝐴 → (𝑓𝑥) ∈ 𝐽))
1910, 18jaod 860 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) → (𝑓𝑥) ∈ 𝐽))
201, 19syl5 34 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 ∈ {∅, 𝐴} → (𝑓𝑥) ∈ 𝐽))
2120ralrimiv 3145 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)
2221ex 412 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽))
2322pm4.71d 561 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
24 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
25 elmapg 8879 . . . 4 ((𝐴𝑉𝑋𝐽) → (𝑓 ∈ (𝐴m 𝑋) ↔ 𝑓:𝑋𝐴))
2624, 13, 25syl2anr 597 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐴m 𝑋) ↔ 𝑓:𝑋𝐴))
27 indistopon 23008 . . . 4 (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28 iscn 23243 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
2927, 28sylan2 593 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
3023, 26, 293bitr4rd 312 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ 𝑓 ∈ (𝐴m 𝑋)))
3130eqrdv 2735 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴m 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  c0 4333  {cpr 4628  ccnv 5684  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Topctop 22899  TopOnctopon 22916   Cn ccn 23232
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235
This theorem is referenced by:  indishmph  23806  indistgp  24108  indispconn  35239
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