Theorem resdmres 5025

Description: Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
resdmres	|- ( A |` dom ( A |` B ) ) = ( A |` B )

Proof of Theorem resdmres

Step	Hyp	Ref	Expression
1		in12 3282	. . . 4 |- ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) = ( ( B X. _V ) i^i ( A i^i ( dom A X. _V ) ) )
2		df-res 4546	. . . . . 6 |- ( A |` dom A ) = ( A i^i ( dom A X. _V ) )
3		resdm2 5024	. . . . . 6 |- ( A |` dom A ) = `' `' A
4	2, 3	eqtr3i 2160	. . . . 5 |- ( A i^i ( dom A X. _V ) ) = `' `' A
5	4	ineq2i 3269	. . . 4 |- ( ( B X. _V ) i^i ( A i^i ( dom A X. _V ) ) ) = ( ( B X. _V ) i^i `' `' A )
6		incom 3263	. . . 4 |- ( ( B X. _V ) i^i `' `' A ) = ( `' `' A i^i ( B X. _V ) )
7	1, 5, 6	3eqtri 2162	. . 3 |- ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) = ( `' `' A i^i ( B X. _V ) )
8		df-res 4546	. . . 4 |- ( A |` dom ( A |` B ) ) = ( A i^i ( dom ( A |` B ) X. _V ) )
9		dmres 4835	. . . . . . 7 |- dom ( A |` B ) = ( B i^i dom A )
10	9	xpeq1i 4554	. . . . . 6 |- ( dom ( A |` B ) X. _V ) = ( ( B i^i dom A ) X. _V )
11		xpindir 4670	. . . . . 6 |- ( ( B i^i dom A ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
12	10, 11	eqtri 2158	. . . . 5 |- ( dom ( A |` B ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
13	12	ineq2i 3269	. . . 4 |- ( A i^i ( dom ( A |` B ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) )
14	8, 13	eqtri 2158	. . 3 |- ( A |` dom ( A |` B ) ) = ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) )
15		df-res 4546	. . 3 |- ( `' `' A |` B ) = ( `' `' A i^i ( B X. _V ) )
16	7, 14, 15	3eqtr4i 2168	. 2 |- ( A |` dom ( A |` B ) ) = ( `' `' A |` B )
17		rescnvcnv 4996	. 2 |- ( `' `' A |` B ) = ( A |` B )
18	16, 17	eqtri 2158	1 |- ( A |` dom ( A |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1331   _Vcvv 2681   i^i cin 3065   X. cxp 4532   `'ccnv 4533   dom cdm 4534   |` cres 4536

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546

This theorem is referenced by:  imadmres  5026  metreslem  12538