Theorem resdmres 5122

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 3348	. . . 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 4640	. . . . . 6 |- ( A |` dom A ) = ( A i^i ( dom A X. _V ) )
3		resdm2 5121	. . . . . 6 |- ( A |` dom A ) = `' `' A
4	2, 3	eqtr3i 2200	. . . . 5 |- ( A i^i ( dom A X. _V ) ) = `' `' A
5	4	ineq2i 3335	. . . 4 |- ( ( B X. _V ) i^i ( A i^i ( dom A X. _V ) ) ) = ( ( B X. _V ) i^i `' `' A )
6		incom 3329	. . . 4 |- ( ( B X. _V ) i^i `' `' A ) = ( `' `' A i^i ( B X. _V ) )
7	1, 5, 6	3eqtri 2202	. . 3 |- ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) = ( `' `' A i^i ( B X. _V ) )
8		df-res 4640	. . . 4 |- ( A |` dom ( A |` B ) ) = ( A i^i ( dom ( A |` B ) X. _V ) )
9		dmres 4930	. . . . . . 7 |- dom ( A |` B ) = ( B i^i dom A )
10	9	xpeq1i 4648	. . . . . 6 |- ( dom ( A |` B ) X. _V ) = ( ( B i^i dom A ) X. _V )
11		xpindir 4765	. . . . . 6 |- ( ( B i^i dom A ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
12	10, 11	eqtri 2198	. . . . 5 |- ( dom ( A |` B ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
13	12	ineq2i 3335	. . . 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 2198	. . 3 |- ( A |` dom ( A |` B ) ) = ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) )
15		df-res 4640	. . 3 |- ( `' `' A |` B ) = ( `' `' A i^i ( B X. _V ) )
16	7, 14, 15	3eqtr4i 2208	. 2 |- ( A |` dom ( A |` B ) ) = ( `' `' A |` B )
17		rescnvcnv 5093	. 2 |- ( `' `' A |` B ) = ( A |` B )
18	16, 17	eqtri 2198	1 |- ( A |` dom ( A |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1353   _Vcvv 2739   i^i cin 3130   X. cxp 4626   `'ccnv 4627   dom cdm 4628   |` cres 4630

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640

This theorem is referenced by:  imadmres  5123  metreslem  13965