Theorem rnhmpha 14889

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 4191	. 2 |- (A ~= B -> B e. ran ~= )
4		rnhmph 14887	. . 3 |- ran ~= C_ Top
5	4	sseli 2617	. 2 |- (B e. ran ~= -> B e. Top)
6	3, 5	syl 12	1 |- (A ~= B -> B e. Top)

Syntax hints:   -> wi 3   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  ran crn 3987  Topctop 8857   ~= chomeo 10231

This theorem is referenced by:  hmpher 14890

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-hmph 10233

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