Theorem hmph 10447

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 1531	. . . 4 |- (j = J -> (j e. Top <-> J e. Top))
2		opreq1 3959	. . . . . 6 |- (j = J -> (j Homeo k) = (J Homeo k))
3	2	eleq2d 1538	. . . . 5 |- (j = J -> (f e. (j Homeo k) <-> f e. (J Homeo k)))
4	3	exbidv 1277	. . . 4 |- (j = J -> (E.f f e. (j Homeo k) <-> E.f f e. (J Homeo k)))
5	1, 4	3anbi13d 893	. . 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 1531	. . . . . 6 |- (k = K -> (k e. Top <-> K e. Top))
7	6	bicomd 520	. . . . 5 |- (k = K -> (K e. Top <-> k e. Top))
8		opreq2 3960	. . . . . . . 8 |- (K = k -> (J Homeo K) = (J Homeo k))
9	8	eqcoms 1475	. . . . . . 7 |- (k = K -> (J Homeo K) = (J Homeo k))
10	9	eleq2d 1538	. . . . . 6 |- (k = K -> (f e. (J Homeo K) <-> f e. (J Homeo k)))
11	10	exbidv 1277	. . . . 5 |- (k = K -> (E.f f e. (J Homeo K) <-> E.f f e. (J Homeo k)))
12	7, 11	3anbi23d 894	. . . 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 776	. . . 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 534	. . 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 10446	. . 3 |- ~= = {<.j, k>. | (j e. Top /\ k e. Top /\ E.f f e. (j Homeo k))}
16	5, 14, 15	brabg 2813	. 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 652	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 774   = wceq 954   e. wcel 956  E.wex 978   class class class wbr 2614  (class class class)co 3954  Topctop 7538   Homeo chomeosm 10436   ~= chomeo 10437

This theorem is referenced by:  hmphsyma 10451  hmphre 10453  hmphtr 10454  homcard 10462

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956  df-hmph 10446

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