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Theorem cnindis 21006
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 4168 . . . . . . 7 (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2 topontop 20641 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32ad2antrr 761 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝐽 ∈ Top)
4 0opn 20634 . . . . . . . . . 10 (𝐽 ∈ Top → ∅ ∈ 𝐽)
53, 4syl 17 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∅ ∈ 𝐽)
6 imaeq2 5421 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑓𝑥) = (𝑓 “ ∅))
7 ima0 5440 . . . . . . . . . . 11 (𝑓 “ ∅) = ∅
86, 7syl6eq 2671 . . . . . . . . . 10 (𝑥 = ∅ → (𝑓𝑥) = ∅)
98eleq1d 2683 . . . . . . . . 9 (𝑥 = ∅ → ((𝑓𝑥) ∈ 𝐽 ↔ ∅ ∈ 𝐽))
105, 9syl5ibrcom 237 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = ∅ → (𝑓𝑥) ∈ 𝐽))
11 fimacnv 6303 . . . . . . . . . . 11 (𝑓:𝑋𝐴 → (𝑓𝐴) = 𝑋)
1211adantl 482 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) = 𝑋)
13 toponmax 20643 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1413ad2antrr 761 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝑋𝐽)
1512, 14eqeltrd 2698 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) ∈ 𝐽)
16 imaeq2 5421 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑓𝑥) = (𝑓𝐴))
1716eleq1d 2683 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑓𝑥) ∈ 𝐽 ↔ (𝑓𝐴) ∈ 𝐽))
1815, 17syl5ibrcom 237 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = 𝐴 → (𝑓𝑥) ∈ 𝐽))
1910, 18jaod 395 . . . . . . 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 2959 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)
2221ex 450 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽))
2322pm4.71d 665 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
24 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
25 elmapg 7815 . . . 4 ((𝐴𝑉𝑋𝐽) → (𝑓 ∈ (𝐴𝑚 𝑋) ↔ 𝑓:𝑋𝐴))
2624, 13, 25syl2anr 495 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐴𝑚 𝑋) ↔ 𝑓:𝑋𝐴))
27 indistopon 20715 . . . 4 (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28 iscn 20949 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
2927, 28sylan2 491 . . 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 2619 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴𝑚 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wral 2907  c0 3891  {cpr 4150  ccnv 5073  cima 5077  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  Topctop 20617  TopOnctopon 20618   Cn ccn 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-top 20621  df-topon 20623  df-cn 20941
This theorem is referenced by:  indishmph  21511  indistgp  21814  indispconn  30921
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