Theorem topgrpbs 14974

Description: In a topological group, the base set of the topology and the base set of the group are equal.

Hypotheses

Ref	Expression
topgrpi.1	|- G = (1st` K)
topgrpi.2	|- J = (2nd` K)

Assertion

Ref	Expression
topgrpbs	|- (K e. TopGrp -> ran G = U.J)

Proof of Theorem topgrpbs

Step	Hyp	Ref	Expression
1		topgrpi.1	. . 3 |- G = (1st` K)
2		topgrpi.2	. . 3 |- J = (2nd` K)
3		eqid 1884	. . 3 |- (inv` G) = (inv` G)
4	1, 2, 3	topgrpi 14972	. 2 |- (K e. TopGrp -> ((G e. Grp /\ J e. Top) /\ G e. ((J X.t J) Cn J) /\ (inv` G) e. (J Cn J)))
5		eqid 1884	. . . . . 6 |- ran G = ran G
6	5, 3	grpinvf 9364	. . . . 5 |- (G e. Grp -> (inv` G):ran G-1-1-onto->ran G)
7		f1of 4635	. . . . . . . . . 10 |- ((inv` G):ran G-1-1-onto->ran G -> (inv` G):ran G-->ran G)
8		fdm 4567	. . . . . . . . . . 11 |- ((inv` G):ran G-->ran G -> dom (inv` G) = ran G)
9		eqtr2 1905	. . . . . . . . . . . . 13 |- ((dom (inv` G) = ran G /\ dom (inv` G) = U.J) -> ran G = U.J)
10	9	ex 402	. . . . . . . . . . . 12 |- (dom (inv` G) = ran G -> (dom (inv` G) = U.J -> ran G = U.J))
11		fdm 4567	. . . . . . . . . . . 12 |- ((inv` G):U.J-->U.J -> dom (inv` G) = U.J)
12	10, 11	syl5 20	. . . . . . . . . . 11 |- (dom (inv` G) = ran G -> ((inv` G):U.J-->U.J -> ran G = U.J))
13	8, 12	syl 12	. . . . . . . . . 10 |- ((inv` G):ran G-->ran G -> ((inv` G):U.J-->U.J -> ran G = U.J))
14	7, 13	syl 12	. . . . . . . . 9 |- ((inv` G):ran G-1-1-onto->ran G -> ((inv` G):U.J-->U.J -> ran G = U.J))
15		eqid 1884	. . . . . . . . . 10 |- U.J = U.J
16	15, 15	cnf 9038	. . . . . . . . 9 |- ((J e. Top /\ J e. Top /\ (inv` G) e. (J Cn J)) -> (inv` G):U.J-->U.J)
17	14, 16	syl5com 63	. . . . . . . 8 |- ((J e. Top /\ J e. Top /\ (inv` G) e. (J Cn J)) -> ((inv` G):ran G-1-1-onto->ran G -> ran G = U.J))
18	17	3exp 1066	. . . . . . 7 |- (J e. Top -> (J e. Top -> ((inv` G) e. (J Cn J) -> ((inv` G):ran G-1-1-onto->ran G -> ran G = U.J))))
19	18	pm2.43i 78	. . . . . 6 |- (J e. Top -> ((inv` G) e. (J Cn J) -> ((inv` G):ran G-1-1-onto->ran G -> ran G = U.J)))
20	19	com3r 39	. . . . 5 |- ((inv` G):ran G-1-1-onto->ran G -> (J e. Top -> ((inv` G) e. (J Cn J) -> ran G = U.J)))
21	6, 20	syl 12	. . . 4 |- (G e. Grp -> (J e. Top -> ((inv` G) e. (J Cn J) -> ran G = U.J)))
22	21	imp31 389	. . 3 |- (((G e. Grp /\ J e. Top) /\ (inv` G) e. (J Cn J)) -> ran G = U.J)
23	22	3adant2 895	. 2 |- (((G e. Grp /\ J e. Top) /\ G e. ((J X.t J) Cn J) /\ (inv` G) e. (J Cn J)) -> ran G = U.J)
24	4, 23	syl 12	1 |- (K e. TopGrp -> ran G = U.J)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  U.cuni 3177  dom cdm 3986  ran crn 3987  -->wf 3994  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Topctop 8857   X.t ctx 8930   Cn ccn 9028  Grpcgr 9311  invcgn 9313  TopGrpctopgrp 14969

This theorem is referenced by:  topgrpsubcnlem 14981  trhom 14983  tpgprop1 14986  tpgprop2 14987

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-reu 2111  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-if 2983  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-map 5383  df-cn 9030  df-grp 9316  df-gid 9317  df-ginv 9318  df-topgrp 14970

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