Theorem iscnp3 14371

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 14367	. 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 5406	. . . . . . . . . 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 14194	. . . . . . . . . . 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 5409	. . . . . . . . . . 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 3218	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> x C_ dom F )
9		funimass3 5674	. . . . . . . . 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 2491	. . . . . 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 2494	. . . 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 1020	. 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 980   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   C_ wss 3153   `'ccnv 4658   dom cdm 4659   "cima 4662   Fun wfun 5248   -->wf 5250   ` cfv 5254  (class class class)co 5918  TopOnctopon 14178   CnP ccnp 14354

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-top 14166  df-topon 14179  df-cnp 14357

This theorem is referenced by:  cncnpi  14396  cnpdis  14410