Theorem rnhmpha 10521

Description: The relation "being homeomorph to" implies the operands are topologies.

Hypotheses

Ref	Expression
rnhmpha.1	|- A e. V
rnhmpha.2	|- B e. V

Assertion

Ref	Expression
rnhmpha	|- (A ~= B -> B e. Top)

Proof of Theorem rnhmpha

Step	Hyp	Ref	Expression
1		rnhmpha.1	. . 3 |- A e. V
2		rnhmpha.2	. . 3 |- B e. V
3	1, 2	brelrn 3350	. 2 |- (A ~= B -> B e. ran ~= )
4		rnhmph 10519	. . 3 |- ran ~= (_ Top
5	4	sseli 2068	. 2 |- (B e. ran ~= -> B e. Top)
6	3, 5	syl 10	1 |- (A ~= B -> B e. Top)

Syntax hints:   -> wi 3   e. wcel 960  Vcvv 1814   class class class wbr 2624  ran crn 3177  Topctop 7590   ~= chomeo 10500

This theorem is referenced by:  hmpher 10522

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195  df-hmph 10509

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