Theorem idcncf 15117

Description: The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 15113 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 3214	. . 3 ⊢ ℂ ⊆ ℂ
3		cncfmptid 15113	. . 3 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ))
4	2, 2, 3	mp2an 426	. 2 ⊢ (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ)
5	1, 4	eqeltri 2279	1 ⊢ 𝐹 ∈ (ℂ–cn→ℂ)

Syntax hints:   = wceq 1373   ∈ wcel 2177   ⊆ wss 3167   ↦ cmpt 4109  (class class class)co 5951  ℂcc 7930  –cn→ccncf 15086

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-map 6744  df-cncf 15087

This theorem is referenced by:  sub1cncf  15118  sub2cncf  15119