Theorem cncfmptid 14168

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 3165	. 2 |- ( ( S C_ T /\ T C_ CC ) -> S C_ CC )
2		simpr 110	. 2 |- ( ( S C_ T /\ T C_ CC ) -> T C_ CC )
3		simpll 527	. . . . 5 |- ( ( ( S C_ T /\ T C_ CC ) /\ x e. S ) -> S C_ T )
4		simpr 110	. . . . 5 |- ( ( ( S C_ T /\ T C_ CC ) /\ x e. S ) -> x e. S )
5	3, 4	sseldd 3158	. . . 4 |- ( ( ( S C_ T /\ T C_ CC ) /\ x e. S ) -> x e. T )
6	5	fmpttd 5673	. . 3 |- ( ( S C_ T /\ T C_ CC ) -> ( x e. S |-> x ) : S --> T )
7		simpr 110	. . . 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 2177	. . . . . . . 8 |- ( x e. S |-> x ) = ( x e. S |-> x )
10		id 19	. . . . . . . 8 |- ( x = y -> x = y )
11		simprll 537	. . . . . . . 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 5611	. . . . . . 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 538	. . . . . . . 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 5611	. . . . . . 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 5895	. . . . . 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 5521	. . . . 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 4015	. . . 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 382	. . 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 14151	. 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 433	1 |- ( ( S C_ T /\ T C_ CC ) -> ( x e. S |-> x ) e. ( S -cn-> T ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   C_ wss 3131   class class class wbr 4005   |-> cmpt 4066   ` cfv 5218  (class class class)co 5877   CCcc 7811   < clt 7994   - cmin 8130   RR+crp 9655   abscabs 11008   -cn->ccncf 14142

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-map 6652  df-cncf 14143

This theorem is referenced by:  expcncf  14177  dvcnp2cntop  14248