Theorem cncfi 13215

Description: Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)

Assertion

Ref	Expression
cncfi	|- ( ( F e. ( A -cn-> B ) /\ C e. A /\ R e. RR+ ) -> E. z e. RR+ A. w e. A ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) )

Distinct variable groups:   z, w, A   w, C, z   w, F, z   w, R, z   w, B, z

Proof of Theorem cncfi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cncfrss 13212	. . . . . 6 |- ( F e. ( A -cn-> B ) -> A C_ CC )
2		cncfrss2 13213	. . . . . 6 |- ( F e. ( A -cn-> B ) -> B C_ CC )
3		elcncf2 13211	. . . . . 6 |- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) ) )
4	1, 2, 3	syl2anc 409	. . . . 5 |- ( F e. ( A -cn-> B ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) ) )
5	4	ibi 175	. . . 4 |- ( F e. ( A -cn-> B ) -> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) )
6	5	simprd 113	. . 3 |- ( F e. ( A -cn-> B ) -> A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) )
7		oveq2 5850	. . . . . . . 8 |- ( x = C -> ( w - x ) = ( w - C ) )
8	7	fveq2d 5490	. . . . . . 7 |- ( x = C -> ( abs ` ( w - x ) ) = ( abs ` ( w - C ) ) )
9	8	breq1d 3992	. . . . . 6 |- ( x = C -> ( ( abs ` ( w - x ) ) < z <-> ( abs ` ( w - C ) ) < z ) )
10		fveq2 5486	. . . . . . . . 9 |- ( x = C -> ( F ` x ) = ( F ` C ) )
11	10	oveq2d 5858	. . . . . . . 8 |- ( x = C -> ( ( F ` w ) - ( F ` x ) ) = ( ( F ` w ) - ( F ` C ) ) )
12	11	fveq2d 5490	. . . . . . 7 |- ( x = C -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) = ( abs ` ( ( F ` w ) - ( F ` C ) ) ) )
13	12	breq1d 3992	. . . . . 6 |- ( x = C -> ( ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y <-> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < y ) )
14	9, 13	imbi12d 233	. . . . 5 |- ( x = C -> ( ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) <-> ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < y ) ) )
15	14	rexralbidv 2492	. . . 4 |- ( x = C -> ( E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) <-> E. z e. RR+ A. w e. A ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < y ) ) )
16		breq2 3986	. . . . . 6 |- ( y = R -> ( ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < y <-> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) )
17	16	imbi2d 229	. . . . 5 |- ( y = R -> ( ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < y ) <-> ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) ) )
18	17	rexralbidv 2492	. . . 4 |- ( y = R -> ( E. z e. RR+ A. w e. A ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < y ) <-> E. z e. RR+ A. w e. A ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) ) )
19	15, 18	rspc2v 2843	. . 3 |- ( ( C e. A /\ R e. RR+ ) -> ( A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) -> E. z e. RR+ A. w e. A ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) ) )
20	6, 19	mpan9 279	. 2 |- ( ( F e. ( A -cn-> B ) /\ ( C e. A /\ R e. RR+ ) ) -> E. z e. RR+ A. w e. A ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) )
21	20	3impb 1189	1 |- ( ( F e. ( A -cn-> B ) /\ C e. A /\ R e. RR+ ) -> E. z e. RR+ A. w e. A ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   C_ wss 3116   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   CCcc 7751   < clt 7933   - cmin 8069   RR+crp 9589   abscabs 10939   -cn->ccncf 13207

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-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

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-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  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-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  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-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-map 6616  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-2 8916  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-cncf 13208

This theorem is referenced by:  cncffvrn  13219  climcncf  13221  cncfco  13228  mulcncf  13241  ivthinclemlopn  13264  ivthinclemuopn  13266  eflt  13346