Theorem idcncf 15702

Description: The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 15698 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 3268	. . 3 |- CC C_ CC
3		cncfmptid 15698	. . 3 |- ( ( CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> x ) e. ( CC -cn-> CC ) )
4	2, 2, 3	mp2an 430	. 2 |- ( x e. CC |-> x ) e. ( CC -cn-> CC )
5	1, 4	eqeltri 2311	1 |- F e. ( CC -cn-> CC )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   C_ wss 3220   |-> cmpt 4192  (class class class)co 6085   CCcc 8177   -cn->ccncf 15671

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-map 6924  df-cncf 15672

This theorem is used by:  sub1cncf  15703  sub2cncf  15704