Theorem resdmres 5226

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 3416	. . . 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 4735	. . . . . 6 |- ( A |` dom A ) = ( A i^i ( dom A X. _V ) )
3		resdm2 5225	. . . . . 6 |- ( A |` dom A ) = `' `' A
4	2, 3	eqtr3i 2252	. . . . 5 |- ( A i^i ( dom A X. _V ) ) = `' `' A
5	4	ineq2i 3403	. . . 4 |- ( ( B X. _V ) i^i ( A i^i ( dom A X. _V ) ) ) = ( ( B X. _V ) i^i `' `' A )
6		incom 3397	. . . 4 |- ( ( B X. _V ) i^i `' `' A ) = ( `' `' A i^i ( B X. _V ) )
7	1, 5, 6	3eqtri 2254	. . 3 |- ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) = ( `' `' A i^i ( B X. _V ) )
8		df-res 4735	. . . 4 |- ( A |` dom ( A |` B ) ) = ( A i^i ( dom ( A |` B ) X. _V ) )
9		dmres 5032	. . . . . . 7 |- dom ( A |` B ) = ( B i^i dom A )
10	9	xpeq1i 4743	. . . . . 6 |- ( dom ( A |` B ) X. _V ) = ( ( B i^i dom A ) X. _V )
11		xpindir 4864	. . . . . 6 |- ( ( B i^i dom A ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
12	10, 11	eqtri 2250	. . . . 5 |- ( dom ( A |` B ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
13	12	ineq2i 3403	. . . 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 2250	. . 3 |- ( A |` dom ( A |` B ) ) = ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) )
15		df-res 4735	. . 3 |- ( `' `' A |` B ) = ( `' `' A i^i ( B X. _V ) )
16	7, 14, 15	3eqtr4i 2260	. 2 |- ( A |` dom ( A |` B ) ) = ( `' `' A |` B )
17		rescnvcnv 5197	. 2 |- ( `' `' A |` B ) = ( A |` B )
18	16, 17	eqtri 2250	1 |- ( A |` dom ( A |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1395   _Vcvv 2800   i^i cin 3197   X. cxp 4721   `'ccnv 4722   dom cdm 4723   |` cres 4725

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735

This theorem is referenced by:  imadmres  5227  metreslem  15094