Theorem metreslem 12751

Description: Lemma for metres 12754. (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 5076	. 2 |- ( D |` dom ( D |` ( R X. R ) ) ) = ( D |` ( R X. R ) )
2		ineq2 3302	. . . 4 |- ( dom D = ( X X. X ) -> ( ( R X. R ) i^i dom D ) = ( ( R X. R ) i^i ( X X. X ) ) )
3		dmres 4886	. . . 4 |- dom ( D |` ( R X. R ) ) = ( ( R X. R ) i^i dom D )
4		inxp 4719	. . . . 5 |- ( ( X X. X ) i^i ( R X. R ) ) = ( ( X i^i R ) X. ( X i^i R ) )
5		incom 3299	. . . . 5 |- ( ( X X. X ) i^i ( R X. R ) ) = ( ( R X. R ) i^i ( X X. X ) )
6	4, 5	eqtr3i 2180	. . . 4 |- ( ( X i^i R ) X. ( X i^i R ) ) = ( ( R X. R ) i^i ( X X. X ) )
7	2, 3, 6	3eqtr4g 2215	. . 3 |- ( dom D = ( X X. X ) -> dom ( D |` ( R X. R ) ) = ( ( X i^i R ) X. ( X i^i R ) ) )
8	7	reseq2d 4865	. 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 2204	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 1335   i^i cin 3101   X. cxp 4583   dom cdm 4585   |` cres 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4591  df-rel 4592  df-cnv 4593  df-dm 4595  df-rn 4596  df-res 4597

This theorem is referenced by:  xmetres  12753  metres  12754