Theorem cncfmptid 13223

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 3150	. 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 519	. . . . 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 3143	. . . 4 |- ( ( ( S C_ T /\ T C_ CC ) /\ x e. S ) -> x e. T )
6	5	fmpttd 5640	. . 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 2165	. . . . . . . 8 |- ( x e. S |-> x ) = ( x e. S |-> x )
10		id 19	. . . . . . . 8 |- ( x = y -> x = y )
11		simprll 527	. . . . . . . 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 5579	. . . . . . 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 528	. . . . . . . 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 5579	. . . . . . 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 5860	. . . . . 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 5490	. . . . 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 3992	. . . 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 380	. . 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 13206	. 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 430	1 |- ( ( S C_ T /\ T C_ CC ) -> ( x e. S |-> x ) e. ( S -cn-> T ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   C_ wss 3116   class class class wbr 3982   |-> cmpt 4043   ` cfv 5188  (class class class)co 5842   CCcc 7751   < clt 7933   - cmin 8069   RR+crp 9589   abscabs 10939   -cn->ccncf 13197

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-map 6616  df-cncf 13198

This theorem is referenced by:  expcncf  13232  dvcnp2cntop  13303