Theorem cncfi 15301

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 15298	. . . . . 6 |- ( F e. ( A -cn-> B ) -> A C_ CC )
2		cncfrss2 15299	. . . . . 6 |- ( F e. ( A -cn-> B ) -> B C_ CC )
3		elcncf2 15297	. . . . . 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 6025	. . . . . . . 8 |- ( x = C -> ( w - x ) = ( w - C ) )
8	7	fveq2d 5643	. . . . . . 7 |- ( x = C -> ( abs ` ( w - x ) ) = ( abs ` ( w - C ) ) )
9	8	breq1d 4098	. . . . . 6 |- ( x = C -> ( ( abs ` ( w - x ) ) < z <-> ( abs ` ( w - C ) ) < z ) )
10		fveq2 5639	. . . . . . . . 9 |- ( x = C -> ( F ` x ) = ( F ` C ) )
11	10	oveq2d 6033	. . . . . . . 8 |- ( x = C -> ( ( F ` w ) - ( F ` x ) ) = ( ( F ` w ) - ( F ` C ) ) )
12	11	fveq2d 5643	. . . . . . 7 |- ( x = C -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) = ( abs ` ( ( F ` w ) - ( F ` C ) ) ) )
13	12	breq1d 4098	. . . . . 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 2558	. . . 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 4092	. . . . . 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 2558	. . . 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 2923	. . 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 1225	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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   C_ wss 3200   class class class wbr 4088   -->wf 5322   ` cfv 5326  (class class class)co 6017   CCcc 8029   < clt 8213   - cmin 8349   RR+crp 9887   abscabs 11557   -cn->ccncf 15293

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-map 6818  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-2 9201  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-cncf 15294

This theorem is referenced by:  cncfcdm  15305  climcncf  15307  cncfco  15314  mulcncf  15331  ivthinclemlopn  15359  ivthinclemuopn  15361  eflt  15498