Theorem rnhmph 10533

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 10523	. . . . 5 |- ~= = {<.x, y>. | (x e. Top /\ y e. Top /\ E.z z e. (x Homeo y))}
2		df-3an 777	. . . . . 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 2671	. . . . 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 1495	. . . 4 |- ~= = {<.x, y>. | ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y))}
5		opabssxp 3234	. . . 4 |- {<.x, y>. | ((x e. Top /\ y e. Top) /\ E.z z e. (x Homeo y))} (_ (Top X. Top)
6	4, 5	eqsstr 2091	. . 3 |- ~= (_ (Top X. Top)
7		rnss 3342	. . 3 |- ( ~= (_ (Top X. Top) -> ran ~= (_ ran (Top X. Top))
8	6, 7	ax-mp 7	. 2 |- ran ~= (_ ran (Top X. Top)
9		rnxpss 3474	. 2 |- ran (Top X. Top) (_ Top
10	8, 9	sstri 2073	1 |- ran ~= (_ Top

Syntax hints:   /\ wa 223   /\ w3a 775   e. wcel 958  E.wex 980   (_ wss 2047  {copab 2666   X. cxp 3168  ran crn 3171  (class class class)co 3963  Topctop 7588   Homeo chomeosm 10513   ~= chomeo 10514

This theorem is referenced by:  rnhmpha 10535

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-hmph 10523

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