Theorem idcncf 15324

Description: The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 15320 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 e. CC |-> x )

Assertion

Ref	Expression
idcncf	|- F e. ( CC -cn-> CC )

Proof of Theorem idcncf

Step	Hyp	Ref	Expression
1		idcncf.1	. 2 |- F = ( x e. CC |-> x )
2		ssid 3247	. . 3 |- CC C_ CC
3		cncfmptid 15320	. . 3 |- ( ( CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> x ) e. ( CC -cn-> CC ) )
4	2, 2, 3	mp2an 426	. 2 |- ( x e. CC |-> x ) e. ( CC -cn-> CC )
5	1, 4	eqeltri 2304	1 |- F e. ( CC -cn-> CC )

Syntax hints:   = wceq 1397   e. wcel 2202   C_ wss 3200   |-> cmpt 4150  (class class class)co 6017   CCcc 8029   -cn->ccncf 15293

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-map 6818  df-cncf 15294

This theorem is referenced by:  sub1cncf  15325  sub2cncf  15326