Theorem iscnp3 14917

Description: The predicate "the class F is a continuous function from topology J to topology K at point P". (Contributed by NM, 15-May-2007.)

Assertion

Ref	Expression
iscnp3	|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) ) )

Distinct variable groups:   x, y, F   x, J, y   x, K, y   x, X, y   x, Y, y   x, P, y

Proof of Theorem iscnp3

Step	Hyp	Ref	Expression
1		iscnp 14913	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) )
2		ffun 5482	. . . . . . . . . 10 |- ( F : X --> Y -> Fun F )
3	2	ad2antlr 489	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> Fun F )
4		toponss 14740	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ x e. J ) -> x C_ X )
5	4	adantlr 477	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> x C_ X )
6		fdm 5485	. . . . . . . . . . 11 |- ( F : X --> Y -> dom F = X )
7	6	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> dom F = X )
8	5, 7	sseqtrrd 3264	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> x C_ dom F )
9		funimass3 5759	. . . . . . . . 9 |- ( ( Fun F /\ x C_ dom F ) -> ( ( F " x ) C_ y <-> x C_ ( `' F " y ) ) )
10	3, 8, 9	syl2anc 411	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> ( ( F " x ) C_ y <-> x C_ ( `' F " y ) ) )
11	10	anbi2d 464	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> ( ( P e. x /\ ( F " x ) C_ y ) <-> ( P e. x /\ x C_ ( `' F " y ) ) ) )
12	11	rexbidva 2527	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ F : X --> Y ) -> ( E. x e. J ( P e. x /\ ( F " x ) C_ y ) <-> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) )
13	12	imbi2d 230	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ F : X --> Y ) -> ( ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) <-> ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) )
14	13	ralbidv 2530	. . . 4 |- ( ( J e. (TopOn ` X ) /\ F : X --> Y ) -> ( A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) <-> A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) )
15	14	pm5.32da 452	. . 3 |- ( J e. (TopOn ` X ) -> ( ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) ) )
16	15	3ad2ant1 1042	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) ) )
17	1, 16	bitrd 188	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   C_ wss 3198   `'ccnv 4722   dom cdm 4723   "cima 4726   Fun wfun 5318   -->wf 5320   ` cfv 5324  (class class class)co 6013  TopOnctopon 14724   CnP ccnp 14900

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-map 6814  df-top 14712  df-topon 14725  df-cnp 14903

This theorem is referenced by:  cncnpi  14942  cnpdis  14956