Theorem cncfi 15443

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 15440	. . . . . 6 |- ( F e. ( A -cn-> B ) -> A C_ CC )
2		cncfrss2 15441	. . . . . 6 |- ( F e. ( A -cn-> B ) -> B C_ CC )
3		elcncf2 15439	. . . . . 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 411	. . . . 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 176	. . . 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 114	. . 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 6058	. . . . . . . 8 |- ( x = C -> ( w - x ) = ( w - C ) )
8	7	fveq2d 5674	. . . . . . 7 |- ( x = C -> ( abs ` ( w - x ) ) = ( abs ` ( w - C ) ) )
9	8	breq1d 4119	. . . . . 6 |- ( x = C -> ( ( abs ` ( w - x ) ) < z <-> ( abs ` ( w - C ) ) < z ) )
10		fveq2 5670	. . . . . . . . 9 |- ( x = C -> ( F ` x ) = ( F ` C ) )
11	10	oveq2d 6066	. . . . . . . 8 |- ( x = C -> ( ( F ` w ) - ( F ` x ) ) = ( ( F ` w ) - ( F ` C ) ) )
12	11	fveq2d 5674	. . . . . . 7 |- ( x = C -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) = ( abs ` ( ( F ` w ) - ( F ` C ) ) ) )
13	12	breq1d 4119	. . . . . 6 |- ( x = C -> ( ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y <-> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < y ) )
14	9, 13	imbi12d 234	. . . . 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 2568	. . . 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 4113	. . . . . 6 |- ( y = R -> ( ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < y <-> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) )
17	16	imbi2d 230	. . . . 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 2568	. . . 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 2934	. . 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 281	. 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 1226	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 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   C_ wss 3211   class class class wbr 4109   -->wf 5348   ` cfv 5352  (class class class)co 6050   CCcc 8125   < clt 8308   - cmin 8444   RR+crp 9986   abscabs 11682   -cn->ccncf 15435

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-map 6884  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-2 9296  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-cncf 15436

This theorem is referenced by:  cncfcdm  15447  climcncf  15449  cncfco  15456  mulcncf  15473  ivthinclemlopn  15501  ivthinclemuopn  15503  eflt  15640