Theorem iscnp3 15194

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 15190	. 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 5516	. . . . . . . . . 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 15017	. . . . . . . . . . 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 5519	. . . . . . . . . . 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 3281	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> x C_ dom F )
9		funimass3 5799	. . . . . . . . 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 2541	. . . . . 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 2544	. . . 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 1045	. 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 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   C_ wss 3214   `'ccnv 4753   dom cdm 4754   "cima 4757   Fun wfun 5351   -->wf 5353   ` cfv 5357  (class class class)co 6058  TopOnctopon 15001   CnP ccnp 15177

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-top 14989  df-topon 15002  df-cnp 15180

This theorem is referenced by:  cncnpi  15219  cnpdis  15233