Theorem cncnp2 7779

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 7762	. . . . . 6 |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> F:X-->Y)
4	3	3expia 835	. . . . 5 |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) -> F:X-->Y))
5	4	pm4.71rd 639	. . . 4 |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ F e. (J Cn K))))
6	1, 2	cncnp 7778	. . . . . 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 833	. . . . 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 649	. . . 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 528	. . 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 799	. 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 7763	. . . . . . . 8 |- (((J e. Top /\ K e. Top /\ x e. X) /\ F e. ((J CnP K)` x)) -> F:X-->Y)
12	11	ex 373	. . . . . . 7 |- ((J e. Top /\ K e. Top /\ x e. X) -> (F e. ((J CnP K)` x) -> F:X-->Y))
13	12	3expa 833	. . . . . 6 |- (((J e. Top /\ K e. Top) /\ x e. X) -> (F e. ((J CnP K)` x) -> F:X-->Y))
14	13	pm4.71rd 639	. . . . 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 1659	. . . 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 2350	. . . 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 540	. . 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 828	. 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 531	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 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  (/)c0 2280  U.cuni 2503  -->wf 3178  ` cfv 3182  (class class class)co 3963  Topctop 7588   Cn ccn 7752   CnP ccnp 7753

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324  df-top 7592  df-cn 7754  df-cnp 7755

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