Theorem idhmeo 22924

Description: The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
idhmeo	⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽Homeo𝐽))

Proof of Theorem idhmeo

Step	Hyp	Ref	Expression
1		idcn 22408	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
2		cnvresid 6513	. . 3 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
3	2, 1	eqeltrid 2843	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ◡( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
4		ishmeo 22910	. 2 ⊢ (( I ↾ 𝑋) ∈ (𝐽Homeo𝐽) ↔ (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ ◡( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)))
5	1, 3, 4	sylanbrc 583	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽Homeo𝐽))

Syntax hints:   → wi 4   ∈ wcel 2106   I cid 5488  ^◡ccnv 5588   ↾ cres 5591  ‘cfv 6433  (class class class)co 7275  TopOnctopon 22059   Cn ccn 22375  Homeochmeo 22904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-top 22043  df-topon 22060  df-cn 22378  df-hmeo 22906

This theorem is referenced by:  hmphref  22932