Theorem cnmptid 15305

Description: The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypothesis

Ref	Expression
cnmptid.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))

Assertion

Ref	Expression
cnmptid	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋

Proof of Theorem cnmptid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equcom 1758	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2	1	opabbii 4193	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
3		df-id 4433	. . . . 5 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4		mptv 4223	. . . . 5 ⊢ (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
5	2, 3, 4	3eqtr4i 2269	. . . 4 ⊢ I = (𝑥 ∈ V ↦ 𝑥)
6	5	reseq1i 5054	. . 3 ⊢ ( I ↾ 𝑋) = ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋)
7		ssv 3270	. . . 4 ⊢ 𝑋 ⊆ V
8		resmpt 5106	. . . 4 ⊢ (𝑋 ⊆ V → ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥))
9	7, 8	ax-mp 5	. . 3 ⊢ ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥)
10	6, 9	eqtri 2259	. 2 ⊢ ( I ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥)
11		cnmptid.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
12		idcn 15236	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
13	11, 12	syl 14	. 2 ⊢ (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
14	10, 13	eqeltrrid 2326	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  {copab 4186   ↦ cmpt 4187   I cid 4428   ↾ cres 4771  ‘cfv 5372  (class class class)co 6075  TopOnctopon 15034   Cn ccn 15209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-top 15022  df-topon 15035  df-cn 15212

This theorem is referenced by:  imasnopn  15323  expcn  15593