Theorem cncfmptid 12752

Description: The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)

Assertion

Ref	Expression
cncfmptid	|- ( ( S C_ T /\ T C_ CC ) -> ( x e. S |-> x ) e. ( S -cn-> T ) )

Distinct variable groups:   x, S   x, T

Proof of Theorem cncfmptid

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr 3105	. 2 |- ( ( S C_ T /\ T C_ CC ) -> S C_ CC )
2		simpr 109	. 2 |- ( ( S C_ T /\ T C_ CC ) -> T C_ CC )
3		simpll 518	. . . . 5 |- ( ( ( S C_ T /\ T C_ CC ) /\ x e. S ) -> S C_ T )
4		simpr 109	. . . . 5 |- ( ( ( S C_ T /\ T C_ CC ) /\ x e. S ) -> x e. S )
5	3, 4	sseldd 3098	. . . 4 |- ( ( ( S C_ T /\ T C_ CC ) /\ x e. S ) -> x e. T )
6	5	fmpttd 5575	. . 3 |- ( ( S C_ T /\ T C_ CC ) -> ( x e. S |-> x ) : S --> T )
7		simpr 109	. . . 4 |- ( ( y e. S /\ w e. RR+ ) -> w e. RR+ )
8	7	a1i 9	. . 3 |- ( ( S C_ T /\ T C_ CC ) -> ( ( y e. S /\ w e. RR+ ) -> w e. RR+ ) )
9		eqid 2139	. . . . . . . 8 |- ( x e. S |-> x ) = ( x e. S |-> x )
10		id 19	. . . . . . . 8 |- ( x = y -> x = y )
11		simprll 526	. . . . . . . 8 |- ( ( ( S C_ T /\ T C_ CC ) /\ ( ( y e. S /\ z e. S ) /\ w e. RR+ ) ) -> y e. S )
12	9, 10, 11, 11	fvmptd3 5514	. . . . . . 7 |- ( ( ( S C_ T /\ T C_ CC ) /\ ( ( y e. S /\ z e. S ) /\ w e. RR+ ) ) -> ( ( x e. S |-> x ) ` y ) = y )
13		id 19	. . . . . . . 8 |- ( x = z -> x = z )
14		simprlr 527	. . . . . . . 8 |- ( ( ( S C_ T /\ T C_ CC ) /\ ( ( y e. S /\ z e. S ) /\ w e. RR+ ) ) -> z e. S )
15	9, 13, 14, 14	fvmptd3 5514	. . . . . . 7 |- ( ( ( S C_ T /\ T C_ CC ) /\ ( ( y e. S /\ z e. S ) /\ w e. RR+ ) ) -> ( ( x e. S |-> x ) ` z ) = z )
16	12, 15	oveq12d 5792	. . . . . 6 |- ( ( ( S C_ T /\ T C_ CC ) /\ ( ( y e. S /\ z e. S ) /\ w e. RR+ ) ) -> ( ( ( x e. S |-> x ) ` y ) - ( ( x e. S |-> x ) ` z ) ) = ( y - z ) )
17	16	fveq2d 5425	. . . . 5 |- ( ( ( S C_ T /\ T C_ CC ) /\ ( ( y e. S /\ z e. S ) /\ w e. RR+ ) ) -> ( abs ` ( ( ( x e. S |-> x ) ` y ) - ( ( x e. S |-> x ) ` z ) ) ) = ( abs ` ( y - z ) ) )
18	17	breq1d 3939	. . . 4 |- ( ( ( S C_ T /\ T C_ CC ) /\ ( ( y e. S /\ z e. S ) /\ w e. RR+ ) ) -> ( ( abs ` ( ( ( x e. S |-> x ) ` y ) - ( ( x e. S |-> x ) ` z ) ) ) < w <-> ( abs ` ( y - z ) ) < w ) )
19	18	exbiri 379	. . 3 |- ( ( S C_ T /\ T C_ CC ) -> ( ( ( y e. S /\ z e. S ) /\ w e. RR+ ) -> ( ( abs ` ( y - z ) ) < w -> ( abs ` ( ( ( x e. S |-> x ) ` y ) - ( ( x e. S |-> x ) ` z ) ) ) < w ) ) )
20	6, 8, 19	elcncf1di 12735	. 2 |- ( ( S C_ T /\ T C_ CC ) -> ( ( S C_ CC /\ T C_ CC ) -> ( x e. S |-> x ) e. ( S -cn-> T ) ) )
21	1, 2, 20	mp2and 429	1 |- ( ( S C_ T /\ T C_ CC ) -> ( x e. S |-> x ) e. ( S -cn-> T ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   C_ wss 3071   class class class wbr 3929   |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   CCcc 7618   < clt 7800   - cmin 7933   RR+crp 9441   abscabs 10769   -cn->ccncf 12726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-map 6544  df-cncf 12727

This theorem is referenced by:  expcncf  12761  dvcnp2cntop  12832