Theorem cnpf2 12212

Description: A continuous function at point P is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)

Assertion

Ref	Expression
cnpf2	|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> F : X --> Y )

Proof of Theorem cnpf2

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 964	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> F e. ( ( J CnP K ) ` P ) )
2		topontop 12018	. . . . 5 |- ( K e. (TopOn ` Y ) -> K e. Top )
3		cnprcl2k 12211	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ K e. Top /\ F e. ( ( J CnP K ) ` P ) ) -> P e. X )
4	2, 3	syl3an2 1231	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> P e. X )
5		iscnp 12204	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. a e. K ( ( F ` P ) e. a -> E. b e. J ( P e. b /\ ( F " b ) C_ a ) ) ) ) )
6	4, 5	syld3an3 1242	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. a e. K ( ( F ` P ) e. a -> E. b e. J ( P e. b /\ ( F " b ) C_ a ) ) ) ) )
7	1, 6	mpbid 146	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> ( F : X --> Y /\ A. a e. K ( ( F ` P ) e. a -> E. b e. J ( P e. b /\ ( F " b ) C_ a ) ) ) )
8	7	simpld 111	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> F : X --> Y )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 943   e. wcel 1461   A.wral 2388   E.wrex 2389   C_ wss 3035   "cima 4500   -->wf 5075   ` cfv 5079  (class class class)co 5726   Topctop 12001  TopOnctopon 12014   CnP ccnp 12192

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-map 6496  df-top 12002  df-topon 12015  df-cnp 12195

This theorem is referenced by:  iscnp4  12223  cnptopco  12227  cncnp2m  12236  cnptopresti  12243  lmtopcnp  12255  txcnp  12276  metcnpi3  12500  cnplimcim  12586  limccnpcntop  12594  limccnp2lem  12595  limccnp2cntop  12596