Theorem hmph 10411

Description: Express the predicate J is homeomorph to K.

Assertion

Ref	Expression
hmph	|- ((J e. Top /\ K e. Top) -> (J ~= K <-> E.f f e. (J Homeo K)))

Distinct variable groups:   f,J   f,K

Proof of Theorem hmph

Step	Hyp	Ref	Expression
1		eleq1 1526	. . . 4 |- (j = J -> (j e. Top <-> J e. Top))
2		opreq1 3953	. . . . . 6 |- (j = J -> (j Homeo k) = (J Homeo k))
3	2	eleq2d 1533	. . . . 5 |- (j = J -> (f e. (j Homeo k) <-> f e. (J Homeo k)))
4	3	exbidv 1274	. . . 4 |- (j = J -> (E.f f e. (j Homeo k) <-> E.f f e. (J Homeo k)))
5	1, 4	3anbi13d 892	. . 3 |- (j = J -> ((j e. Top /\ k e. Top /\ E.f f e. (j Homeo k)) <-> (J e. Top /\ k e. Top /\ E.f f e. (J Homeo k))))
6		eleq1 1526	. . . . . 6 |- (k = K -> (k e. Top <-> K e. Top))
7	6	bicomd 519	. . . . 5 |- (k = K -> (K e. Top <-> k e. Top))
8		opreq2 3954	. . . . . . . 8 |- (K = k -> (J Homeo K) = (J Homeo k))
9	8	eqcoms 1470	. . . . . . 7 |- (k = K -> (J Homeo K) = (J Homeo k))
10	9	eleq2d 1533	. . . . . 6 |- (k = K -> (f e. (J Homeo K) <-> f e. (J Homeo k)))
11	10	exbidv 1274	. . . . 5 |- (k = K -> (E.f f e. (J Homeo K) <-> E.f f e. (J Homeo k)))
12	7, 11	3anbi23d 893	. . . 4 |- (k = K -> ((J e. Top /\ K e. Top /\ E.f f e. (J Homeo K)) <-> (J e. Top /\ k e. Top /\ E.f f e. (J Homeo k))))
13		df-3an 775	. . . 4 |- ((J e. Top /\ K e. Top /\ E.f f e. (J Homeo K)) <-> ((J e. Top /\ K e. Top) /\ E.f f e. (J Homeo K)))
14	12, 13	syl5rbbr 533	. . 3 |- (k = K -> ((J e. Top /\ k e. Top /\ E.f f e. (J Homeo k)) <-> ((J e. Top /\ K e. Top) /\ E.f f e. (J Homeo K))))
15		df-hmph 10410	. . 3 |- ~= = {<.j, k>. | (j e. Top /\ k e. Top /\ E.f f e. (j Homeo k))}
16	5, 14, 15	brabg 2807	. 2 |- ((J e. Top /\ K e. Top) -> (J ~= K <-> ((J e. Top /\ K e. Top) /\ E.f f e. (J Homeo K))))
17	16	bianabs 651	1 |- ((J e. Top /\ K e. Top) -> (J ~= K <-> E.f f e. (J Homeo K)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  E.wex 977   class class class wbr 2609  (class class class)co 3948  Topctop 7530   Homeo chomeosm 10400   ~= chomeo 10401

This theorem is referenced by:  hmphsyma 10415  hmphre 10417  hmphtr 10418  homcard 10426

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188  df-opr 3950  df-hmph 10410

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