Theorem metreslem 22167

Description: Lemma for metres 22170. (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 5625	. 2 |- ( D |` dom ( D |` ( R X. R ) ) ) = ( D |` ( R X. R ) )
2		ineq2 3808	. . . 4 |- ( dom D = ( X X. X ) -> ( ( R X. R ) i^i dom D ) = ( ( R X. R ) i^i ( X X. X ) ) )
3		dmres 5419	. . . 4 |- dom ( D |` ( R X. R ) ) = ( ( R X. R ) i^i dom D )
4		inxp 5254	. . . . 5 |- ( ( X X. X ) i^i ( R X. R ) ) = ( ( X i^i R ) X. ( X i^i R ) )
5		incom 3805	. . . . 5 |- ( ( X X. X ) i^i ( R X. R ) ) = ( ( R X. R ) i^i ( X X. X ) )
6	4, 5	eqtr3i 2646	. . . 4 |- ( ( X i^i R ) X. ( X i^i R ) ) = ( ( R X. R ) i^i ( X X. X ) )
7	2, 3, 6	3eqtr4g 2681	. . 3 |- ( dom D = ( X X. X ) -> dom ( D |` ( R X. R ) ) = ( ( X i^i R ) X. ( X i^i R ) ) )
8	7	reseq2d 5396	. 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	syl5eqr 2670	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 1483   i^i cin 3573   X. cxp 5112   dom cdm 5114   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  xmetres  22169  metres  22170