Theorem cncfmptid 13462

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 3156	. 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 525	. . . . 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 3149	. . . 4 |- ( ( ( S C_ T /\ T C_ CC ) /\ x e. S ) -> x e. T )
6	5	fmpttd 5655	. . 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 2171	. . . . . . . 8 |- ( x e. S |-> x ) = ( x e. S |-> x )
10		id 19	. . . . . . . 8 |- ( x = y -> x = y )
11		simprll 533	. . . . . . . 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 5593	. . . . . . 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 534	. . . . . . . 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 5593	. . . . . . 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 5875	. . . . . 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 5503	. . . . 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 4000	. . . 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 13445	. 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 431	1 |- ( ( S C_ T /\ T C_ CC ) -> ( x e. S |-> x ) e. ( S -cn-> T ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2142   C_ wss 3122   class class class wbr 3990   |-> cmpt 4051   ` cfv 5200  (class class class)co 5857   CCcc 7776   < clt 7958   - cmin 8094   RR+crp 9614   abscabs 10965   -cn->ccncf 13436

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-cnex 7869

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-ral 2454  df-rex 2455  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-fv 5208  df-ov 5860  df-oprab 5861  df-mpo 5862  df-map 6632  df-cncf 13437

This theorem is referenced by:  expcncf  13471  dvcnp2cntop  13542