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Theorem ssidcn 23158
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 23138 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
2 f1oi 6877 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1of 6839 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋𝑋
54biantrur 530 . . 3 (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
61, 5bitr4di 289 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
7 cnvresid 6632 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
87imaeq1i 6060 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
9 elssuni 4940 . . . . . . . . 9 (𝑥𝐾𝑥 𝐾)
109adantl 481 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥 𝐾)
11 toponuni 22815 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = 𝐾)
1211ad2antlr 726 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑋 = 𝐾)
1310, 12sseqtrrd 4021 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥𝑋)
14 resiima 6079 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1513, 14syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
168, 15eqtrid 2780 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1716eleq1d 2814 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
1817ralbidva 3172 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ ∀𝑥𝐾 𝑥𝐽))
19 dfss3 3968 . . 3 (𝐾𝐽 ↔ ∀𝑥𝐾 𝑥𝐽)
2018, 19bitr4di 289 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝐾𝐽))
216, 20bitrd 279 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3058  wss 3947   cuni 4908   I cid 5575  ccnv 5677  cres 5680  cima 5681  wf 6544  1-1-ontowf1o 6547  cfv 6548  (class class class)co 7420  TopOnctopon 22811   Cn ccn 23127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8846  df-top 22795  df-topon 22812  df-cn 23130
This theorem is referenced by:  idcn  23160  sshauslem  23275
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