Theorem mapdiscn 10511

Description: Any mapping whose domain is associated to the discrete topology is continuous.

Hypotheses

Ref	Expression
mapdiscn.1	|- F e. C
mapdiscn.2	|- B = U.J
mapdiscn.3	|- A e. V

Assertion

Ref	Expression
mapdiscn	|- ((J e. Top /\ F:A-->B) -> F e. (P~A Cn J))

Proof of Theorem mapdiscn

Step	Hyp	Ref	Expression
1		mapdiscn.3	. . 3 |- A e. V
2	1	distop 7649	. 2 |- P~A e. Top
3		unipw 2756	. . . . . . . 8 |- U.P~A = A
4	3	eqcomi 1479	. . . . . . 7 |- A = U.P~A
5		mapdiscn.2	. . . . . . 7 |- B = U.J
6		feq23 3623	. . . . . . 7 |- ((A = U.P~A /\ B = U.J) -> (F:A-->B <-> F:U.P~A-->U.J))
7	4, 5, 6	mp2an 697	. . . . . 6 |- (F:A-->B <-> F:U.P~A-->U.J)
8	7	biimp 151	. . . . 5 |- (F:A-->B -> F:U.P~A-->U.J)
9	8	adantl 388	. . . 4 |- (((P~A e. Top /\ J e. Top) /\ F:A-->B) -> F:U.P~A-->U.J)
10		imassrn 3415	. . . . . . 7 |- (`'F"x) (_ ran `' F
11		dfdm4 3305	. . . . . . . . . 10 |- dom F = ran `' F
12	11	eqcomi 1479	. . . . . . . . 9 |- ran `' F = dom F
13	12	sseq2i 2086	. . . . . . . 8 |- ((`'F"x) (_ ran `' F <-> (`'F"x) (_ dom F)
14		fdm 3631	. . . . . . . . . . 11 |- (F:A-->B -> dom F = A)
15		sseq2 2083	. . . . . . . . . . . . 13 |- (dom F = A -> ((`'F"x) (_ dom F <-> (`'F"x) (_ A))
16	15	biimpd 153	. . . . . . . . . . . 12 |- (dom F = A -> ((`'F"x) (_ dom F -> (`'F"x) (_ A))
17		fex 3652	. . . . . . . . . . . . . . . 16 |- ((F:A-->B /\ A e. V) -> F e. V)
18	1, 17	mpan2 696	. . . . . . . . . . . . . . 15 |- (F:A-->B -> F e. V)
19		cnvexg 3519	. . . . . . . . . . . . . . 15 |- (F e. V -> `'F e. V)
20	18, 19	syl 10	. . . . . . . . . . . . . 14 |- (F:A-->B -> `'F e. V)
21		imaexg 3416	. . . . . . . . . . . . . 14 |- (`'F e. V -> (`'F"x) e. V)
22		elpwg 2405	. . . . . . . . . . . . . 14 |- ((`'F"x) e. V -> ((`'F"x) e. P~A <-> (`'F"x) (_ A))
23	20, 21, 22	3syl 20	. . . . . . . . . . . . 13 |- (F:A-->B -> ((`'F"x) e. P~A <-> (`'F"x) (_ A))
24	23	biimprd 154	. . . . . . . . . . . 12 |- (F:A-->B -> ((`'F"x) (_ A -> (`'F"x) e. P~A))
25	16, 24	syl9 57	. . . . . . . . . . 11 |- (dom F = A -> (F:A-->B -> ((`'F"x) (_ dom F -> (`'F"x) e. P~A)))
26	14, 25	mpcom 49	. . . . . . . . . 10 |- (F:A-->B -> ((`'F"x) (_ dom F -> (`'F"x) e. P~A))
27	26	ad2antlr 405	. . . . . . . . 9 |- ((((P~A e. Top /\ J e. Top) /\ F:A-->B) /\ x e. J) -> ((`'F"x) (_ dom F -> (`'F"x) e. P~A))
28	27	com12 11	. . . . . . . 8 |- ((`'F"x) (_ dom F -> ((((P~A e. Top /\ J e. Top) /\ F:A-->B) /\ x e. J) -> (`'F"x) e. P~A))
29	13, 28	sylbi 199	. . . . . . 7 |- ((`'F"x) (_ ran `' F -> ((((P~A e. Top /\ J e. Top) /\ F:A-->B) /\ x e. J) -> (`'F"x) e. P~A))
30	10, 29	ax-mp 7	. . . . . 6 |- ((((P~A e. Top /\ J e. Top) /\ F:A-->B) /\ x e. J) -> (`'F"x) e. P~A)
31	30	ex 373	. . . . 5 |- (((P~A e. Top /\ J e. Top) /\ F:A-->B) -> (x e. J -> (`'F"x) e. P~A))
32	31	r19.21aiv 1713	. . . 4 |- (((P~A e. Top /\ J e. Top) /\ F:A-->B) -> A.x e. J (`'F"x) e. P~A)
33	9, 32	jca 288	. . 3 |- (((P~A e. Top /\ J e. Top) /\ F:A-->B) -> (F:U.P~A-->U.J /\ A.x e. J (`'F"x) e. P~A))
34		eqid 1475	. . . . 5 |- U.P~A = U.P~A
35		eqid 1475	. . . . 5 |- U.J = U.J
36	34, 35	iscn 7758	. . . 4 |- ((P~A e. Top /\ J e. Top) -> (F e. (P~A Cn J) <-> (F:U.P~A-->U.J /\ A.x e. J (`'F"x) e. P~A)))
37	36	adantr 389	. . 3 |- (((P~A e. Top /\ J e. Top) /\ F:A-->B) -> (F e. (P~A Cn J) <-> (F:U.P~A-->U.J /\ A.x e. J (`'F"x) e. P~A)))
38	33, 37	mpbird 196	. 2 |- (((P~A e. Top /\ J e. Top) /\ F:A-->B) -> F e. (P~A Cn J))
39	2, 38	mpanl1 706	1 |- ((J e. Top /\ F:A-->B) -> F e. (P~A Cn J))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   (_ wss 2047  P~cpw 2401  U.cuni 2503  `'ccnv 3169  dom cdm 3170  ran crn 3171  "cima 3173  -->wf 3178  (class class class)co 3963  Topctop 7588   Cn ccn 7752

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-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

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