Theorem hmph 10510

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 1537	. . . 4 |- (j = J -> (j e. Top <-> J e. Top))
2		opreq1 3974	. . . . . 6 |- (j = J -> (j Homeo k) = (J Homeo k))
3	2	eleq2d 1544	. . . . 5 |- (j = J -> (f e. (j Homeo k) <-> f e. (J Homeo k)))
4	3	exbidv 1281	. . . 4 |- (j = J -> (E.f f e. (j Homeo k) <-> E.f f e. (J Homeo k)))
5	1, 4	3anbi13d 897	. . 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 1537	. . . . . 6 |- (k = K -> (k e. Top <-> K e. Top))
7	6	bicomd 523	. . . . 5 |- (k = K -> (K e. Top <-> k e. Top))
8		opreq2 3975	. . . . . . . 8 |- (K = k -> (J Homeo K) = (J Homeo k))
9	8	eqcoms 1481	. . . . . . 7 |- (k = K -> (J Homeo K) = (J Homeo k))
10	9	eleq2d 1544	. . . . . 6 |- (k = K -> (f e. (J Homeo K) <-> f e. (J Homeo k)))
11	10	exbidv 1281	. . . . 5 |- (k = K -> (E.f f e. (J Homeo K) <-> E.f f e. (J Homeo k)))
12	7, 11	3anbi23d 898	. . . 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 779	. . . 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 537	. . 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 10509	. . 3 |- ~= = {<.j, k>. | (j e. Top /\ k e. Top /\ E.f f e. (j Homeo k))}
16	5, 14, 15	brabg 2824	. 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 655	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 777   = wceq 958   e. wcel 960  E.wex 982   class class class wbr 2624  (class class class)co 3969  Topctop 7590   Homeo chomeosm 10499   ~= chomeo 10500

This theorem is referenced by:  hmphsyma 10514  hmphre 10516  hmphtr 10517  homcard 10525  homcardus 10526  eqindhome 10527

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971  df-hmph 10509

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