Theorem cnpimaex 21060

Description: Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)

Assertion

Ref	Expression
cnpimaex	|- ( ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) )

Distinct variable groups:   x, A   x, F   x, J   x, K   x, P

Proof of Theorem cnpimaex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- U. J = U. J
2		eqid 2622	. . . . . 6 |- U. K = U. K
3	1, 2	iscnp2 21043	. . . . 5 |- ( F e. ( ( J CnP K ) ` P ) <-> ( ( J e. Top /\ K e. Top /\ P e. U. J ) /\ ( F : U. J --> U. K /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) )
4	3	simprbi 480	. . . 4 |- ( F e. ( ( J CnP K ) ` P ) -> ( F : U. J --> U. K /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) )
5	4	simprd 479	. . 3 |- ( F e. ( ( J CnP K ) ` P ) -> A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) )
6		eleq2 2690	. . . . 5 |- ( y = A -> ( ( F ` P ) e. y <-> ( F ` P ) e. A ) )
7		sseq2 3627	. . . . . . 7 |- ( y = A -> ( ( F " x ) C_ y <-> ( F " x ) C_ A ) )
8	7	anbi2d 740	. . . . . 6 |- ( y = A -> ( ( P e. x /\ ( F " x ) C_ y ) <-> ( P e. x /\ ( F " x ) C_ A ) ) )
9	8	rexbidv 3052	. . . . 5 |- ( y = A -> ( E. x e. J ( P e. x /\ ( F " x ) C_ y ) <-> E. x e. J ( P e. x /\ ( F " x ) C_ A ) ) )
10	6, 9	imbi12d 334	. . . 4 |- ( y = A -> ( ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) <-> ( ( F ` P ) e. A -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) ) ) )
11	10	rspccv 3306	. . 3 |- ( A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) -> ( A e. K -> ( ( F ` P ) e. A -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) ) ) )
12	5, 11	syl 17	. 2 |- ( F e. ( ( J CnP K ) ` P ) -> ( A e. K -> ( ( F ` P ) e. A -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) ) ) )
13	12	3imp 1256	1 |- ( ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   U.cuni 4436   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698   CnP ccnp 21029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-top 20699  df-topon 20716  df-cnp 21032

This theorem is referenced by:  iscnp4  21067  cnpnei  21068  cnpco  21071  cncnp  21084  cnpresti  21092  lmcnp  21108  txcnpi  21411  txcnp  21423  ptcnplem  21424  cnpflfi  21803  ghmcnp  21918  xrlimcnp  24695  cnambfre  33458