Theorem cncfmptid 14833

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 3191	. 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 3184	. . . 4 |- ( ( ( S C_ T /\ T C_ CC ) /\ x e. S ) -> x e. T )
6	5	fmpttd 5717	. . 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 2196	. . . . . . . 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 5655	. . . . . . 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 5655	. . . . . . 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 5940	. . . . . 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 5562	. . . . 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 4043	. . . 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 14815	. 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 2167   C_ wss 3157   class class class wbr 4033   |-> cmpt 4094   ` cfv 5258  (class class class)co 5922   CCcc 7877   < clt 8061   - cmin 8197   RR+crp 9728   abscabs 11162   -cn->ccncf 14806

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-map 6709  df-cncf 14807

This theorem is referenced by:  idcncf  14837  expcncf  14845  hovercncf  14882  dvcnp2cntop  14935