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Theorem cnindis 21307
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 4392 . . . . . . 7 (𝑥 ∈ {∅, 𝐴} → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
2 topontop 20928 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32ad2antrr 708 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝐽 ∈ Top)
4 0opn 20919 . . . . . . . . . 10 (𝐽 ∈ Top → ∅ ∈ 𝐽)
53, 4syl 17 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∅ ∈ 𝐽)
6 imaeq2 5672 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑓𝑥) = (𝑓 “ ∅))
7 ima0 5691 . . . . . . . . . . 11 (𝑓 “ ∅) = ∅
86, 7syl6eq 2856 . . . . . . . . . 10 (𝑥 = ∅ → (𝑓𝑥) = ∅)
98eleq1d 2870 . . . . . . . . 9 (𝑥 = ∅ → ((𝑓𝑥) ∈ 𝐽 ↔ ∅ ∈ 𝐽))
105, 9syl5ibrcom 238 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = ∅ → (𝑓𝑥) ∈ 𝐽))
11 fimacnv 6565 . . . . . . . . . . 11 (𝑓:𝑋𝐴 → (𝑓𝐴) = 𝑋)
1211adantl 469 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) = 𝑋)
13 toponmax 20941 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1413ad2antrr 708 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → 𝑋𝐽)
1512, 14eqeltrd 2885 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑓𝐴) ∈ 𝐽)
16 imaeq2 5672 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑓𝑥) = (𝑓𝐴))
1716eleq1d 2870 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑓𝑥) ∈ 𝐽 ↔ (𝑓𝐴) ∈ 𝐽))
1815, 17syl5ibrcom 238 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → (𝑥 = 𝐴 → (𝑓𝑥) ∈ 𝐽))
1910, 18jaod 877 . . . . . . 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 3153 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) ∧ 𝑓:𝑋𝐴) → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)
2221ex 399 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 → ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽))
2322pm4.71d 553 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓:𝑋𝐴 ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
24 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
25 elmapg 8101 . . . 4 ((𝐴𝑉𝑋𝐽) → (𝑓 ∈ (𝐴𝑚 𝑋) ↔ 𝑓:𝑋𝐴))
2624, 13, 25syl2anr 586 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐴𝑚 𝑋) ↔ 𝑓:𝑋𝐴))
27 indistopon 21016 . . . 4 (𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
28 iscn 21250 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {∅, 𝐴} ∈ (TopOn‘𝐴)) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
2927, 28sylan2 582 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ (𝑓:𝑋𝐴 ∧ ∀𝑥 ∈ {∅, 𝐴} (𝑓𝑥) ∈ 𝐽)))
3023, 26, 293bitr4rd 303 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝑓 ∈ (𝐽 Cn {∅, 𝐴}) ↔ 𝑓 ∈ (𝐴𝑚 𝑋)))
3130eqrdv 2804 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴𝑚 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2156  wral 3096  c0 4116  {cpr 4372  ccnv 5310  cima 5314  wf 6093  cfv 6097  (class class class)co 6870  𝑚 cmap 8088  Topctop 20908  TopOnctopon 20925   Cn ccn 21239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-map 8090  df-top 20909  df-topon 20926  df-cn 21242
This theorem is referenced by:  indishmph  21812  indistgp  22114  indispconn  31537
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