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Theorem cnindis 21144
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 4230 . . . . . . 7 (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2 topontop 20766 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32ad2antrr 762 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝐽 ∈ Top)
4 0opn 20757 . . . . . . . . . 10 (𝐽 ∈ Top → ∅ ∈ 𝐽)
53, 4syl 17 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∅ ∈ 𝐽)
6 imaeq2 5497 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑓𝑥) = (𝑓 “ ∅))
7 ima0 5516 . . . . . . . . . . 11 (𝑓 “ ∅) = ∅
86, 7syl6eq 2701 . . . . . . . . . 10 (𝑥 = ∅ → (𝑓𝑥) = ∅)
98eleq1d 2715 . . . . . . . . 9 (𝑥 = ∅ → ((𝑓𝑥) ∈ 𝐽 ↔ ∅ ∈ 𝐽))
105, 9syl5ibrcom 237 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = ∅ → (𝑓𝑥) ∈ 𝐽))
11 fimacnv 6387 . . . . . . . . . . 11 (𝑓:𝑋𝐴 → (𝑓𝐴) = 𝑋)
1211adantl 481 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) = 𝑋)
13 toponmax 20778 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1413ad2antrr 762 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝑋𝐽)
1512, 14eqeltrd 2730 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) ∈ 𝐽)
16 imaeq2 5497 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑓𝑥) = (𝑓𝐴))
1716eleq1d 2715 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑓𝑥) ∈ 𝐽 ↔ (𝑓𝐴) ∈ 𝐽))
1815, 17syl5ibrcom 237 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = 𝐴 → (𝑓𝑥) ∈ 𝐽))
1910, 18jaod 394 . . . . . . 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 2994 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)
2221ex 449 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽))
2322pm4.71d 667 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
24 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
25 elmapg 7912 . . . 4 ((𝐴𝑉𝑋𝐽) → (𝑓 ∈ (𝐴𝑚 𝑋) ↔ 𝑓:𝑋𝐴))
2624, 13, 25syl2anr 494 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐴𝑚 𝑋) ↔ 𝑓:𝑋𝐴))
27 indistopon 20853 . . . 4 (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28 iscn 21087 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
2927, 28sylan2 490 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
3023, 26, 293bitr4rd 301 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ 𝑓 ∈ (𝐴𝑚 𝑋)))
3130eqrdv 2649 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴𝑚 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  c0 3948  {cpr 4212  ccnv 5142  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Topctop 20746  TopOnctopon 20763   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-top 20747  df-topon 20764  df-cn 21079
This theorem is referenced by:  indishmph  21649  indistgp  21951  indispconn  31342
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