Theorem metres 15407

Description: A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
metres	|- ( D e. ( Met ` X ) -> ( D |` ( R X. R ) ) e. ( Met ` ( X i^i R ) ) )

Proof of Theorem metres

Step	Hyp	Ref	Expression
1		metf 15375	. . 3 |- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
2		fdm 5534	. . 3 |- ( D : ( X X. X ) --> RR -> dom D = ( X X. X ) )
3		metreslem 15404	. . 3 |- ( dom D = ( X X. X ) -> ( D |` ( R X. R ) ) = ( D |` ( ( X i^i R ) X. ( X i^i R ) ) ) )
4	1, 2, 3	3syl 17	. 2 |- ( D e. ( Met ` X ) -> ( D |` ( R X. R ) ) = ( D |` ( ( X i^i R ) X. ( X i^i R ) ) ) )
5		inss1 3451	. . 3 |- ( X i^i R ) C_ X
6		metres2 15405	. . 3 |- ( ( D e. ( Met ` X ) /\ ( X i^i R ) C_ X ) -> ( D |` ( ( X i^i R ) X. ( X i^i R ) ) ) e. ( Met ` ( X i^i R ) ) )
7	5, 6	mpan2 429	. 2 |- ( D e. ( Met ` X ) -> ( D |` ( ( X i^i R ) X. ( X i^i R ) ) ) e. ( Met ` ( X i^i R ) ) )
8	4, 7	eqeltrd 2315	1 |- ( D e. ( Met ` X ) -> ( D |` ( R X. R ) ) e. ( Met ` ( X i^i R ) ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   X. cxp 4767   dom cdm 4769   |` cres 4771   -->wf 5368   ` cfv 5372   RRcr 8168   Metcmet 14846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-rnegex 8278

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-xr 8354  df-xadd 10154  df-xmet 14853  df-met 14854

This theorem is referenced by: (None)