Theorem mapdiscnlem 14870

Description: Lemma for mapdiscn 14871. 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
mapdiscnlem	|- ((J e. Top /\ F:A-->B) -> F e. (~PA Cn J))

Proof of Theorem mapdiscnlem

Step	Hyp	Ref	Expression
1		mapdiscn.3	. . 3 |- A e. _V
2	1	distop 8919	. 2 |- ~PA e. Top
3		unipw 3504	. . . . . . . 8 |- U.~PA = A
4	3	eqcomi 1888	. . . . . . 7 |- A = U.~PA
5		mapdiscn.2	. . . . . . 7 |- B = U.J
6		feq23 4554	. . . . . . 7 |- ((A = U.~PA /\ B = U.J) -> (F:A-->B <-> F:U.~PA-->U.J))
7	4, 5, 6	mp2an 761	. . . . . 6 |- (F:A-->B <-> F:U.~PA-->U.J)
8	7	biimpi 168	. . . . 5 |- (F:A-->B -> F:U.~PA-->U.J)
9	8	adantl 424	. . . 4 |- (((~PA e. Top /\ J e. Top) /\ F:A-->B) -> F:U.~PA-->U.J)
10		imassrn 4278	. . . . . . 7 |- (`'F"x) C_ ran `' F
11		dfdm4 4151	. . . . . . . . . 10 |- dom F = ran `' F
12	11	eqcomi 1888	. . . . . . . . 9 |- ran `' F = dom F
13	12	sseq2i 2642	. . . . . . . 8 |- ((`'F"x) C_ ran `' F <-> (`'F"x) C_ dom F)
14		fdm 4567	. . . . . . . . . . 11 |- (F:A-->B -> dom F = A)
15		sseq2 2639	. . . . . . . . . . . . 13 |- (dom F = A -> ((`'F"x) C_ dom F <-> (`'F"x) C_ A))
16	15	biimpd 170	. . . . . . . . . . . 12 |- (dom F = A -> ((`'F"x) C_ dom F -> (`'F"x) C_ A))
17		fex 4595	. . . . . . . . . . . . . . . 16 |- ((F:A-->B /\ A e. _V) -> F e. _V)
18	1, 17	mpan2 760	. . . . . . . . . . . . . . 15 |- (F:A-->B -> F e. _V)
19		cnvexg 4424	. . . . . . . . . . . . . . 15 |- (F e. _V -> `'F e. _V)
20	18, 19	syl 12	. . . . . . . . . . . . . 14 |- (F:A-->B -> `'F e. _V)
21		imaexg 4279	. . . . . . . . . . . . . 14 |- (`'F e. _V -> (`'F"x) e. _V)
22		elpwg 3038	. . . . . . . . . . . . . 14 |- ((`'F"x) e. _V -> ((`'F"x) e. ~PA <-> (`'F"x) C_ A))
23	20, 21, 22	3syl 24	. . . . . . . . . . . . 13 |- (F:A-->B -> ((`'F"x) e. ~PA <-> (`'F"x) C_ A))
24	23	biimprd 171	. . . . . . . . . . . 12 |- (F:A-->B -> ((`'F"x) C_ A -> (`'F"x) e. ~PA))
25	16, 24	syl9 71	. . . . . . . . . . 11 |- (dom F = A -> (F:A-->B -> ((`'F"x) C_ dom F -> (`'F"x) e. ~PA)))
26	14, 25	mpcom 60	. . . . . . . . . 10 |- (F:A-->B -> ((`'F"x) C_ dom F -> (`'F"x) e. ~PA))
27	26	ad2antlr 441	. . . . . . . . 9 |- ((((~PA e. Top /\ J e. Top) /\ F:A-->B) /\ x e. J) -> ((`'F"x) C_ dom F -> (`'F"x) e. ~PA))
28	27	com12 14	. . . . . . . 8 |- ((`'F"x) C_ dom F -> ((((~PA e. Top /\ J e. Top) /\ F:A-->B) /\ x e. J) -> (`'F"x) e. ~PA))
29	13, 28	sylbi 216	. . . . . . 7 |- ((`'F"x) C_ ran `' F -> ((((~PA e. Top /\ J e. Top) /\ F:A-->B) /\ x e. J) -> (`'F"x) e. ~PA))
30	10, 29	ax-mp 7	. . . . . 6 |- ((((~PA e. Top /\ J e. Top) /\ F:A-->B) /\ x e. J) -> (`'F"x) e. ~PA)
31	30	ex 402	. . . . 5 |- (((~PA e. Top /\ J e. Top) /\ F:A-->B) -> (x e. J -> (`'F"x) e. ~PA))
32	31	r19.21aiv 2175	. . . 4 |- (((~PA e. Top /\ J e. Top) /\ F:A-->B) -> A.x e. J (`'F"x) e. ~PA)
33	9, 32	jca 310	. . 3 |- (((~PA e. Top /\ J e. Top) /\ F:A-->B) -> (F:U.~PA-->U.J /\ A.x e. J (`'F"x) e. ~PA))
34		eqid 1884	. . . . 5 |- U.~PA = U.~PA
35		eqid 1884	. . . . 5 |- U.J = U.J
36	34, 35	iscn 9034	. . . 4 |- ((~PA e. Top /\ J e. Top) -> (F e. (~PA Cn J) <-> (F:U.~PA-->U.J /\ A.x e. J (`'F"x) e. ~PA)))
37	36	adantr 425	. . 3 |- (((~PA e. Top /\ J e. Top) /\ F:A-->B) -> (F e. (~PA Cn J) <-> (F:U.~PA-->U.J /\ A.x e. J (`'F"x) e. ~PA)))
38	33, 37	mpbird 213	. 2 |- (((~PA e. Top /\ J e. Top) /\ F:A-->B) -> F e. (~PA Cn J))
39	2, 38	mpanl1 770	1 |- ((J e. Top /\ F:A-->B) -> F e. (~PA Cn J))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  U.cuni 3177  `'ccnv 3985  dom cdm 3986  ran crn 3987  "cima 3989  -->wf 3994  (class class class)co 4884  Topctop 8857   Cn ccn 9028

This theorem is referenced by:  mapdiscn 14871

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-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

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