Theorem metss 7821

Description: If two metrics are in a subset relationship, so are their base sets.

Hypotheses

Ref	Expression
metss.1	|- X = dom dom C
metss.2	|- Y = dom dom D

Assertion

Ref	Expression
metss	|- (C (_ D -> X (_ Y)

Proof of Theorem metss

Step	Hyp	Ref	Expression
1		dmss 3316	. . 3 |- (C (_ D -> dom C (_ dom D)
2		dmss 3316	. . 3 |- (dom C (_ dom D -> dom dom C (_ dom dom D)
3	1, 2	syl 10	. 2 |- (C (_ D -> dom dom C (_ dom dom D)
4		metss.1	. 2 |- X = dom dom C
5		metss.2	. 2 |- Y = dom dom D
6	3, 4, 5	3sstr4g 2105	1 |- (C (_ D -> X (_ Y)

Syntax hints:   -> wi 3   = wceq 958   (_ wss 2050  dom cdm 3176

This theorem is referenced by:  metcnss 7895  metcnss2 7896

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-10 968  ax-12 970  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-ss 2056  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-dm 3194

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