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Theorem cnindis 21993
 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 4545 . . . . . . 7 (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2 topontop 21614 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32ad2antrr 726 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝐽 ∈ Top)
4 0opn 21605 . . . . . . . . . 10 (𝐽 ∈ Top → ∅ ∈ 𝐽)
53, 4syl 17 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∅ ∈ 𝐽)
6 imaeq2 5898 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑓𝑥) = (𝑓 “ ∅))
7 ima0 5918 . . . . . . . . . . 11 (𝑓 “ ∅) = ∅
86, 7eqtrdi 2810 . . . . . . . . . 10 (𝑥 = ∅ → (𝑓𝑥) = ∅)
98eleq1d 2837 . . . . . . . . 9 (𝑥 = ∅ → ((𝑓𝑥) ∈ 𝐽 ↔ ∅ ∈ 𝐽))
105, 9syl5ibrcom 250 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = ∅ → (𝑓𝑥) ∈ 𝐽))
11 fimacnv 6831 . . . . . . . . . . 11 (𝑓:𝑋𝐴 → (𝑓𝐴) = 𝑋)
1211adantl 486 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) = 𝑋)
13 toponmax 21627 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1413ad2antrr 726 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝑋𝐽)
1512, 14eqeltrd 2853 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) ∈ 𝐽)
16 imaeq2 5898 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑓𝑥) = (𝑓𝐴))
1716eleq1d 2837 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑓𝑥) ∈ 𝐽 ↔ (𝑓𝐴) ∈ 𝐽))
1815, 17syl5ibrcom 250 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = 𝐴 → (𝑓𝑥) ∈ 𝐽))
1910, 18jaod 857 . . . . . . 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 3113 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)
2221ex 417 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽))
2322pm4.71d 566 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
24 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
25 elmapg 8430 . . . 4 ((𝐴𝑉𝑋𝐽) → (𝑓 ∈ (𝐴m 𝑋) ↔ 𝑓:𝑋𝐴))
2624, 13, 25syl2anr 600 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐴m 𝑋) ↔ 𝑓:𝑋𝐴))
27 indistopon 21702 . . . 4 (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28 iscn 21936 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
2927, 28sylan2 596 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
3023, 26, 293bitr4rd 316 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ 𝑓 ∈ (𝐴m 𝑋)))
3130eqrdv 2757 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴m 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∅c0 4226  {cpr 4525  ◡ccnv 5524   “ cima 5528  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151   ↑m cmap 8417  Topctop 21594  TopOnctopon 21611   Cn ccn 21925 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8419  df-top 21595  df-topon 21612  df-cn 21928 This theorem is referenced by:  indishmph  22499  indistgp  22801  indispconn  32713
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