Theorem rnhmph 10420

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

Assertion

Ref	Expression
rnhmph	|- ran ~= (_ Top

Proof of Theorem rnhmph

Step	Hyp	Ref	Expression
1		df-hmph 10410	. . . . 5 |- ~= = {<.x, y>. | (x e. Top /\ y e. Top /\ E.z z e. (x Homeo y))}
2		df-3an 775	. . . . . 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 2661	. . . . 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 1487	. . . 4 |- ~= = {<.x, y>. | ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y))}
5		opabssxp 3224	. . . 4 |- {<.x, y>. | ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y))} (_ (Top X. Top)
6	4, 5	eqsstr 2081	. . 3 |- ~= (_ (Top X. Top)
7		rnss 3331	. . 3 |- ( ~= (_ (Top X. Top) -> ran ~= (_ ran (Top X. Top))
8	6, 7	ax-mp 7	. 2 |- ran ~= (_ ran (Top X. Top)
9		rnxpss 3460	. 2 |- ran (Top X. Top) (_ Top
10	8, 9	sstri 2063	1 |- ran ~= (_ Top

Syntax hints:   /\ wa 223   /\ w3a 773   e. wcel 955  E.wex 977   (_ wss 2037  {copab 2656   X. cxp 3158  ran crn 3161  (class class class)co 3948  Topctop 7530   Homeo chomeosm 10400   ~= chomeo 10401

This theorem is referenced by:  rnhmpha 10422

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-9 962  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-ral 1641  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-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179  df-hmph 10410

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