Theorem idcncf 15354

Description: The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 15350 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 3246	. . 3 ⊢ ℂ ⊆ ℂ
3		cncfmptid 15350	. . 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 2303	1 ⊢ 𝐹 ∈ (ℂ–cn→ℂ)

Syntax hints:   = wceq 1397   ∈ wcel 2201   ⊆ wss 3199   ↦ cmpt 4151  (class class class)co 6023  ℂcc 8035  –cn→ccncf 15323

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-map 6824  df-cncf 15324

This theorem is referenced by:  sub1cncf  15355  sub2cncf  15356