Theorem homindlem3 14900

Description: Homeomorphisms preserve topological indiscretion.

Assertion

Ref	Expression
homindlem3	|- ((K e. Top /\ A =/= (/) /\ A e. B) -> ({(/), A} ~= K -> K = {(/), U.K}))

Proof of Theorem homindlem3

Step	Hyp	Ref	Expression
1		simpl1 879	. . . 4 |- (((K e. Top /\ A =/= (/) /\ A e. B) /\ {(/), A} ~= K) -> K e. Top)
2		elisset 2299	. . . . . . . . . 10 |- (A e. B -> A e. _V)
3		preq2 3099	. . . . . . . . . . . 12 |- (A = if(A e. _V, A, (/)) -> {(/), A} = {(/), if(A e. _V, A, (/))})
4	3	eleq1d 1963	. . . . . . . . . . 11 |- (A = if(A e. _V, A, (/)) -> ({(/), A} e. Top <-> {(/), if(A e. _V, A, (/))} e. Top))
5		indistop 8918	. . . . . . . . . . 11 |- {(/), if(A e. _V, A, (/))} e. Top
6	4, 5	dedth 3011	. . . . . . . . . 10 |- (A e. _V -> {(/), A} e. Top)
7	2, 6	syl 12	. . . . . . . . 9 |- (A e. B -> {(/), A} e. Top)
8		hmphsyma 14882	. . . . . . . . . . 11 |- (({(/), A} e. Top /\ K e. Top) -> ({(/), A} ~= K -> K ~= {(/), A}))
9		homcard 14893	. . . . . . . . . . . 12 |- ((K e. Top /\ {(/), A} e. Top) -> (K ~= {(/), A} -> K ~~ {(/), A}))
10	9	ancoms 484	. . . . . . . . . . 11 |- (({(/), A} e. Top /\ K e. Top) -> (K ~= {(/), A} -> K ~~ {(/), A}))
11	8, 10	syld 30	. . . . . . . . . 10 |- (({(/), A} e. Top /\ K e. Top) -> ({(/), A} ~= K -> K ~~ {(/), A}))
12	11	ex 402	. . . . . . . . 9 |- ({(/), A} e. Top -> (K e. Top -> ({(/), A} ~= K -> K ~~ {(/), A})))
13	7, 12	syl 12	. . . . . . . 8 |- (A e. B -> (K e. Top -> ({(/), A} ~= K -> K ~~ {(/), A})))
14	13	com12 14	. . . . . . 7 |- (K e. Top -> (A e. B -> ({(/), A} ~= K -> K ~~ {(/), A})))
15	14	a1d 15	. . . . . 6 |- (K e. Top -> (A =/= (/) -> (A e. B -> ({(/), A} ~= K -> K ~~ {(/), A}))))
16	15	3imp1 1081	. . . . 5 |- (((K e. Top /\ A =/= (/) /\ A e. B) /\ {(/), A} ~= K) -> K ~~ {(/), A})
17		0ex 3446	. . . . . . . . 9 |- (/) e. _V
18		necom 2094	. . . . . . . . . 10 |- (A =/= (/) <-> (/) =/= A)
19		unpde2eg2 14406	. . . . . . . . . . . 12 |- (((/) e. _V /\ A e. B /\ (/) =/= A) -> {(/), A} ~~ 2o)
20	19	3exp 1066	. . . . . . . . . . 11 |- ((/) e. _V -> (A e. B -> ((/) =/= A -> {(/), A} ~~ 2o)))
21	20	com13 37	. . . . . . . . . 10 |- ((/) =/= A -> (A e. B -> ((/) e. _V -> {(/), A} ~~ 2o)))
22	18, 21	sylbi 216	. . . . . . . . 9 |- (A =/= (/) -> (A e. B -> ((/) e. _V -> {(/), A} ~~ 2o)))
23	17, 22	mpii 56	. . . . . . . 8 |- (A =/= (/) -> (A e. B -> {(/), A} ~~ 2o))
24	23	a1i 8	. . . . . . 7 |- (K e. Top -> (A =/= (/) -> (A e. B -> {(/), A} ~~ 2o)))
25	24	3imp 1061	. . . . . 6 |- ((K e. Top /\ A =/= (/) /\ A e. B) -> {(/), A} ~~ 2o)
26	25	adantr 425	. . . . 5 |- (((K e. Top /\ A =/= (/) /\ A e. B) /\ {(/), A} ~= K) -> {(/), A} ~~ 2o)
27		entr 5473	. . . . 5 |- ((K ~~ {(/), A} /\ {(/), A} ~~ 2o) -> K ~~ 2o)
28	16, 26, 27	syl11anc 524	. . . 4 |- (((K e. Top /\ A =/= (/) /\ A e. B) /\ {(/), A} ~= K) -> K ~~ 2o)
29	1, 28	jca 310	. . 3 |- (((K e. Top /\ A =/= (/) /\ A e. B) /\ {(/), A} ~= K) -> (K e. Top /\ K ~~ 2o))
30	29	ex 402	. 2 |- ((K e. Top /\ A =/= (/) /\ A e. B) -> ({(/), A} ~= K -> (K e. Top /\ K ~~ 2o)))
31		top2ind 14897	. 2 |- ((K e. Top /\ K ~~ 2o) -> K = {(/), U.K})
32	30, 31	syl6 25	1 |- ((K e. Top /\ A =/= (/) /\ A e. B) -> ({(/), A} ~= K -> K = {(/), U.K}))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  (/)c0 2875  ifcif 2982  {cpr 3045  U.cuni 3177   class class class wbr 3338  2oc2o 5173   ~~ cen 5423  Topctop 8857   ~= chomeo 10231

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-3or 859  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-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-1o 5177  df-2o 5178  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-top 8861  df-homeo 10232  df-hmph 10233

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