Theorem metreslem 15262

Description: Lemma for metres 15265. (Contributed by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
metreslem	|- ( dom D = ( X X. X ) -> ( D |` ( R X. R ) ) = ( D |` ( ( X i^i R ) X. ( X i^i R ) ) ) )

Proof of Theorem metreslem

Step	Hyp	Ref	Expression
1		resdmres 5256	. 2 |- ( D |` dom ( D |` ( R X. R ) ) ) = ( D |` ( R X. R ) )
2		ineq2 3418	. . . 4 |- ( dom D = ( X X. X ) -> ( ( R X. R ) i^i dom D ) = ( ( R X. R ) i^i ( X X. X ) ) )
3		dmres 5061	. . . 4 |- dom ( D |` ( R X. R ) ) = ( ( R X. R ) i^i dom D )
4		inxp 4891	. . . . 5 |- ( ( X X. X ) i^i ( R X. R ) ) = ( ( X i^i R ) X. ( X i^i R ) )
5		incom 3413	. . . . 5 |- ( ( X X. X ) i^i ( R X. R ) ) = ( ( R X. R ) i^i ( X X. X ) )
6	4, 5	eqtr3i 2257	. . . 4 |- ( ( X i^i R ) X. ( X i^i R ) ) = ( ( R X. R ) i^i ( X X. X ) )
7	2, 3, 6	3eqtr4g 2292	. . 3 |- ( dom D = ( X X. X ) -> dom ( D |` ( R X. R ) ) = ( ( X i^i R ) X. ( X i^i R ) ) )
8	7	reseq2d 5040	. 2 |- ( dom D = ( X X. X ) -> ( D |` dom ( D |` ( R X. R ) ) ) = ( D |` ( ( X i^i R ) X. ( X i^i R ) ) ) )
9	1, 8	eqtr3id 2281	1 |- ( dom D = ( X X. X ) -> ( D |` ( R X. R ) ) = ( D |` ( ( X i^i R ) X. ( X i^i R ) ) ) )

Syntax hints:   -> wi 4   = wceq 1398   i^i cin 3212   X. cxp 4749   dom cdm 4751   |` cres 4753

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-cnv 4759  df-dm 4761  df-rn 4762  df-res 4763

This theorem is referenced by:  xmetres  15264  metres  15265