Theorem cncfi 12734

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 12731	. . . . . 6 |- ( F e. ( A -cn-> B ) -> A C_ CC )
2		cncfrss2 12732	. . . . . 6 |- ( F e. ( A -cn-> B ) -> B C_ CC )
3		elcncf2 12730	. . . . . 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 408	. . . . 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 5782	. . . . . . . 8 |- ( x = C -> ( w - x ) = ( w - C ) )
8	7	fveq2d 5425	. . . . . . 7 |- ( x = C -> ( abs ` ( w - x ) ) = ( abs ` ( w - C ) ) )
9	8	breq1d 3939	. . . . . 6 |- ( x = C -> ( ( abs ` ( w - x ) ) < z <-> ( abs ` ( w - C ) ) < z ) )
10		fveq2 5421	. . . . . . . . 9 |- ( x = C -> ( F ` x ) = ( F ` C ) )
11	10	oveq2d 5790	. . . . . . . 8 |- ( x = C -> ( ( F ` w ) - ( F ` x ) ) = ( ( F ` w ) - ( F ` C ) ) )
12	11	fveq2d 5425	. . . . . . 7 |- ( x = C -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) = ( abs ` ( ( F ` w ) - ( F ` C ) ) ) )
13	12	breq1d 3939	. . . . . 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 2461	. . . 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 3933	. . . . . 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 2461	. . . 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 2802	. . 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 1177	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 962   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   C_ wss 3071   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7618   < clt 7800   - cmin 7933   RR+crp 9441   abscabs 10769   -cn->ccncf 12726

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-map 6544  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-2 8779  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-cncf 12727

This theorem is referenced by:  cncffvrn  12738  climcncf  12740  cncfco  12747  mulcncf  12760  ivthinclemlopn  12783  ivthinclemuopn  12785