Theorem dmhmph 10455

Description: ~= is a relation whose domain is included in Top.

Assertion

Ref	Expression
dmhmph	|- dom ~= (_ Top

Proof of Theorem dmhmph

Step	Hyp	Ref	Expression
1		df-hmph 10446	. . . . 5 |- ~= = {<.x, y>. | (x e. Top /\ y e. Top /\ E.z z e. (x Homeo y))}
2		df-3an 776	. . . . . 6 |- ((x e. Top /\ y e. Top /\ E.z z e. (x Homeo y)) <-> ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y)))
3	2	opabbii 2666	. . . . 5 |- {<.x, y>. | (x e. Top /\ y e. Top /\ E.z z e. (x Homeo y))} = {<.x, y>. | ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y))}
4	1, 3	eqtr 1492	. . . 4 |- ~= = {<.x, y>. | ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y))}
5		opabssxp 3229	. . . 4 |- {<.x, y>. | ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y))} (_ (Top X. Top)
6	4, 5	eqsstr 2087	. . 3 |- ~= (_ (Top X. Top)
7		dmss 3305	. . 3 |- ( ~= (_ (Top X. Top) -> dom ~= (_ dom (Top X. Top))
8	6, 7	ax-mp 7	. 2 |- dom ~= (_ dom (Top X. Top)
9		dmxpid 3328	. 2 |- dom (Top X. Top) = Top
10	8, 9	sseqtr 2089	1 |- dom ~= (_ Top

Syntax hints:   /\ wa 223   /\ w3a 774   e. wcel 956  E.wex 978   (_ wss 2043  {copab 2661   X. cxp 3163  dom cdm 3165  (class class class)co 3954  Topctop 7538   Homeo chomeosm 10436   ~= chomeo 10437

This theorem is referenced by:  dmhmpha 10457

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-9 963  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-ral 1646  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-br 2615  df-opab 2662  df-xp 3179  df-dm 3183  df-hmph 10446

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