Theorem idcncf 15685

Description: The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 15681 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)

Hypothesis

Ref	Expression
idcncf.1	⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝑥)

Assertion

Ref	Expression
idcncf	⊢ 𝐹 ∈ (ℂ–cn→ℂ)

Proof of Theorem idcncf

Step	Hyp	Ref	Expression
1		idcncf.1	. 2 ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝑥)
2		ssid 3268	. . 3 ⊢ ℂ ⊆ ℂ
3		cncfmptid 15681	. . 3 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ))
4	2, 2, 3	mp2an 430	. 2 ⊢ (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ)
5	1, 4	eqeltri 2311	1 ⊢ 𝐹 ∈ (ℂ–cn→ℂ)

Syntax hints:   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220   ↦ cmpt 4190  (class class class)co 6079  ℂcc 8171  –cn→ccncf 15654

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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-map 6918  df-cncf 15655

This theorem is referenced by:  sub1cncf  15686  sub2cncf  15687