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Theorem ssidcn 21863
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 21843 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
2 f1oi 6643 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1of 6606 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋𝑋
54biantrur 534 . . 3 (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
61, 5syl6bbr 292 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
7 cnvresid 6421 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
87imaeq1i 5913 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
9 elssuni 4854 . . . . . . . . 9 (𝑥𝐾𝑥 𝐾)
109adantl 485 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥 𝐾)
11 toponuni 21522 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = 𝐾)
1211ad2antlr 726 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑋 = 𝐾)
1310, 12sseqtrrd 3994 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → 𝑥𝑋)
14 resiima 5931 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1513, 14syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
168, 15syl5eq 2871 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
1716eleq1d 2900 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥𝐾) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
1817ralbidva 3191 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ ∀𝑥𝐾 𝑥𝐽))
19 dfss3 3941 . . 3 (𝐾𝐽 ↔ ∀𝑥𝐾 𝑥𝐽)
2018, 19syl6bbr 292 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥𝐾 (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝐾𝐽))
216, 20bitrd 282 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  wss 3919   cuni 4824   I cid 5446  ccnv 5541  cres 5544  cima 5545  wf 6339  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  TopOnctopon 21518   Cn ccn 21832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-top 21502  df-topon 21519  df-cn 21835
This theorem is referenced by:  idcn  21865  sshauslem  21980
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