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Theorem ssidcn 12368
Description: The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
ssidcn ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾𝐽))

Proof of Theorem ssidcn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iscn 12355 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
2 f1oi 5398 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1of 5360 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋𝑋
54biantrur 301 . . 3 (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
61, 5syl6bbr 197 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
7 cnvresid 5192 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
87imaeq1i 4873 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
9 elssuni 3759 . . . . . . . . 9 (𝑥𝐾𝑥 𝐾)
109adantl 275 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥 𝐾)
11 toponuni 12171 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = 𝐾)
1211ad2antlr 480 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑋 = 𝐾)
1310, 12sseqtrrd 3131 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥𝑋)
14 resiima 4892 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1513, 14syl 14 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
168, 15syl5eq 2182 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1716eleq1d 2206 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
1817ralbidva 2431 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ ∀𝑥𝐾 𝑥𝐽))
19 dfss3 3082 . . 3 (𝐾𝐽 ↔ ∀𝑥𝐾 𝑥𝐽)
2018, 19syl6bbr 197 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝐾𝐽))
216, 20bitrd 187 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾𝐽))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wral 2414  wss 3066   cuni 3731   I cid 4205  ccnv 4533  cres 4536  cima 4537  wf 5114  1-1-ontowf1o 5117  cfv 5118  (class class class)co 5767  TopOnctopon 12166   Cn ccn 12343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-map 6537  df-top 12154  df-topon 12167  df-cn 12346
This theorem is referenced by:  idcn  12370
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