Theorem metreslem 14700

Description: Lemma for metres 14703. (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 5162	. 2 |- ( D |` dom ( D |` ( R X. R ) ) ) = ( D |` ( R X. R ) )
2		ineq2 3359	. . . 4 |- ( dom D = ( X X. X ) -> ( ( R X. R ) i^i dom D ) = ( ( R X. R ) i^i ( X X. X ) ) )
3		dmres 4968	. . . 4 |- dom ( D |` ( R X. R ) ) = ( ( R X. R ) i^i dom D )
4		inxp 4801	. . . . 5 |- ( ( X X. X ) i^i ( R X. R ) ) = ( ( X i^i R ) X. ( X i^i R ) )
5		incom 3356	. . . . 5 |- ( ( X X. X ) i^i ( R X. R ) ) = ( ( R X. R ) i^i ( X X. X ) )
6	4, 5	eqtr3i 2219	. . . 4 |- ( ( X i^i R ) X. ( X i^i R ) ) = ( ( R X. R ) i^i ( X X. X ) )
7	2, 3, 6	3eqtr4g 2254	. . 3 |- ( dom D = ( X X. X ) -> dom ( D |` ( R X. R ) ) = ( ( X i^i R ) X. ( X i^i R ) ) )
8	7	reseq2d 4947	. 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 2243	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 1364   i^i cin 3156   X. cxp 4662   dom cdm 4664   |` cres 4666

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676

This theorem is referenced by:  xmetres  14702  metres  14703