Theorem iscnp3 14439

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 14435	. 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 5410	. . . . . . . . . 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 14262	. . . . . . . . . . 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 5413	. . . . . . . . . . 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 3222	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> x C_ dom F )
9		funimass3 5678	. . . . . . . . 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 2494	. . . . . 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 2497	. . . 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 2167   A.wral 2475   E.wrex 2476   C_ wss 3157   `'ccnv 4662   dom cdm 4663   "cima 4666   Fun wfun 5252   -->wf 5254   ` cfv 5258  (class class class)co 5922  TopOnctopon 14246   CnP ccnp 14422

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-top 14234  df-topon 14247  df-cnp 14425

This theorem is referenced by:  cncnpi  14464  cnpdis  14478