Theorem iscnp3 14790

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 14786	. 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 5448	. . . . . . . . . 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 14613	. . . . . . . . . . 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 5451	. . . . . . . . . . 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 3240	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ F : X --> Y ) /\ x e. J ) -> x C_ dom F )
9		funimass3 5719	. . . . . . . . 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 2505	. . . . . 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 2508	. . . 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 1021	. 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 981   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   C_ wss 3174   `'ccnv 4692   dom cdm 4693   "cima 4696   Fun wfun 5284   -->wf 5286   ` cfv 5290  (class class class)co 5967  TopOnctopon 14597   CnP ccnp 14773

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-top 14585  df-topon 14598  df-cnp 14776

This theorem is referenced by:  cncnpi  14815  cnpdis  14829