Theorem cncnp2 9056

Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.)

Hypotheses

Ref	Expression
cncnp.1	|- X = U.J
cncnp.2	|- Y = U.K

Assertion

Ref	Expression
cncnp2	|- ((J e. Top /\ K e. Top /\ X =/= (/)) -> (F e. (J Cn K) <-> A.x e. X F e. ((J CnP K)` x)))

Distinct variable groups:   x,J   x,K   x,F   x,X   x,Y

Proof of Theorem cncnp2

Step	Hyp	Ref	Expression
1		cncnp.1	. . . . . . 7 |- X = U.J
2		cncnp.2	. . . . . . 7 |- Y = U.K
3	1, 2	cnf 9038	. . . . . 6 |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> F:X-->Y)
4	3	3expia 1069	. . . . 5 |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) -> F:X-->Y))
5	4	pm4.71rd 701	. . . 4 |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ F e. (J Cn K))))
6	1, 2	cncnp 9055	. . . . . 6 |- ((J e. Top /\ K e. Top /\ F:X-->Y) -> (F e. (J Cn K) <-> A.x e. X F e. ((J CnP K)` x)))
7	6	3expa 1067	. . . . 5 |- (((J e. Top /\ K e. Top) /\ F:X-->Y) -> (F e. (J Cn K) <-> A.x e. X F e. ((J CnP K)` x)))
8	7	pm5.32da 711	. . . 4 |- ((J e. Top /\ K e. Top) -> ((F:X-->Y /\ F e. (J Cn K)) <-> (F:X-->Y /\ A.x e. X F e. ((J CnP K)` x))))
9	5, 8	bitrd 587	. . 3 |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.x e. X F e. ((J CnP K)` x))))
10	9	3adant3 896	. 2 |- ((J e. Top /\ K e. Top /\ X =/= (/)) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.x e. X F e. ((J CnP K)` x))))
11	1, 2	cnpf 9039	. . . . . . . 8 |- (((J e. Top /\ K e. Top /\ x e. X) /\ F e. ((J CnP K)` x)) -> F:X-->Y)
12	11	ex 402	. . . . . . 7 |- ((J e. Top /\ K e. Top /\ x e. X) -> (F e. ((J CnP K)` x) -> F:X-->Y))
13	12	3expa 1067	. . . . . 6 |- (((J e. Top /\ K e. Top) /\ x e. X) -> (F e. ((J CnP K)` x) -> F:X-->Y))
14	13	pm4.71rd 701	. . . . 5 |- (((J e. Top /\ K e. Top) /\ x e. X) -> (F e. ((J CnP K)` x) <-> (F:X-->Y /\ F e. ((J CnP K)` x))))
15	14	ralbidva 2119	. . . 4 |- ((J e. Top /\ K e. Top) -> (A.x e. X F e. ((J CnP K)` x) <-> A.x e. X (F:X-->Y /\ F e. ((J CnP K)` x))))
16		r19.28zv 2964	. . . 4 |- (X =/= (/) -> (A.x e. X (F:X-->Y /\ F e. ((J CnP K)` x)) <-> (F:X-->Y /\ A.x e. X F e. ((J CnP K)` x))))
17	15, 16	sylan9bb 599	. . 3 |- (((J e. Top /\ K e. Top) /\ X =/= (/)) -> (A.x e. X F e. ((J CnP K)` x) <-> (F:X-->Y /\ A.x e. X F e. ((J CnP K)` x))))
18	17	3impa 1062	. 2 |- ((J e. Top /\ K e. Top /\ X =/= (/)) -> (A.x e. X F e. ((J CnP K)` x) <-> (F:X-->Y /\ A.x e. X F e. ((J CnP K)` x))))
19	10, 18	bitr4d 590	1 |- ((J e. Top /\ K e. Top /\ X =/= (/)) -> (F e. (J Cn K) <-> A.x e. X F e. ((J CnP K)` x)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  (/)c0 2875  U.cuni 3177  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857   Cn ccn 9028   CnP ccnp 9029

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-top 8861  df-cn 9030  df-cnp 9031

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