Theorem rnhmph 10566

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 10556	. . . . 5 |- ~= = {<.x, y>. | (x e. Top /\ y e. Top /\ E.z z e. (x Homeo y))}
2		df-3an 781	. . . . . 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 2686	. . . . 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	eqtri 1502	. . . 4 |- ~= = {<.x, y>. | ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y))}
5		opabssxp 3250	. . . 4 |- {<.x, y>. | ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y))} (_ (Top X. Top)
6	4, 5	eqsstri 2102	. . 3 |- ~= (_ (Top X. Top)
7		rnss 3358	. . 3 |- ( ~= (_ (Top X. Top) -> ran ~= (_ ran (Top X. Top))
8	6, 7	ax-mp 7	. 2 |- ran ~= (_ ran (Top X. Top)
9		rnxpss 3490	. 2 |- ran (Top X. Top) (_ Top
10	8, 9	sstri 2084	1 |- ran ~= (_ Top

Syntax hints:   /\ wa 223   /\ w3a 779   e. wcel 962  E.wex 984   (_ wss 2058  {copab 2681   X. cxp 3184  ran crn 3187  (class class class)co 3979  Topctop 7621   Homeo chomeosm 10546   ~= chomeo 10547

This theorem is referenced by:  rnhmpha 10568

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-sep 2718  ax-pow 2758  ax-pr 2795

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-br 2635  df-opab 2682  df-xp 3200  df-rel 3201  df-cnv 3202  df-dm 3204  df-rn 3205  df-hmph 10556

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