Theorem cnpf2 12376

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 983	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> F e. ( ( J CnP K ) ` P ) )
2		topontop 12181	. . . . 5 |- ( K e. (TopOn ` Y ) -> K e. Top )
3		cnprcl2k 12375	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ K e. Top /\ F e. ( ( J CnP K ) ` P ) ) -> P e. X )
4	2, 3	syl3an2 1250	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> P e. X )
5		iscnp 12368	. . . 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 1261	. . 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 962   e. wcel 1480   A.wral 2416   E.wrex 2417   C_ wss 3071   "cima 4542   -->wf 5119   ` cfv 5123  (class class class)co 5774   Topctop 12164  TopOnctopon 12177   CnP ccnp 12355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-top 12165  df-topon 12178  df-cnp 12358

This theorem is referenced by:  iscnp4  12387  cnptopco  12391  cncnp2m  12400  cnptopresti  12407  lmtopcnp  12419  txcnp  12440  metcnpi3  12686  cnplimcim  12805  limccnpcntop  12813  limccnp2lem  12814  limccnp2cntop  12815