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Theorem ssidcn 21791
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 21771 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
2 f1oi 6645 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1of 6608 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋𝑋
54biantrur 531 . . 3 (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
61, 5syl6bbr 290 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
7 cnvresid 6426 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
87imaeq1i 5919 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
9 elssuni 4859 . . . . . . . . 9 (𝑥𝐾𝑥 𝐾)
109adantl 482 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥 𝐾)
11 toponuni 21450 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = 𝐾)
1211ad2antlr 723 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑋 = 𝐾)
1310, 12sseqtrrd 4005 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥𝑋)
14 resiima 5937 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1513, 14syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
168, 15syl5eq 2865 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1716eleq1d 2894 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
1817ralbidva 3193 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ ∀𝑥𝐾 𝑥𝐽))
19 dfss3 3953 . . 3 (𝐾𝐽 ↔ ∀𝑥𝐾 𝑥𝐽)
2018, 19syl6bbr 290 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝐾𝐽))
216, 20bitrd 280 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933   cuni 4830   I cid 5452  ccnv 5547  cres 5550  cima 5551  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  TopOnctopon 21446   Cn ccn 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-top 21430  df-topon 21447  df-cn 21763
This theorem is referenced by:  idcn  21793  sshauslem  21908
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