Theorem resdmres 5337

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 3670	. . . 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 4857	. . . . . 6 |- ( A |` dom A ) = ( A i^i ( dom A X. _V ) )
3		resdm2 5336	. . . . . 6 |- ( A |` dom A ) = `' `' A
4	2, 3	eqtr3i 2451	. . . . 5 |- ( A i^i ( dom A X. _V ) ) = `' `' A
5	4	ineq2i 3658	. . . 4 |- ( ( B X. _V ) i^i ( A i^i ( dom A X. _V ) ) ) = ( ( B X. _V ) i^i `' `' A )
6		incom 3652	. . . 4 |- ( ( B X. _V ) i^i `' `' A ) = ( `' `' A i^i ( B X. _V ) )
7	1, 5, 6	3eqtri 2453	. . 3 |- ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) = ( `' `' A i^i ( B X. _V ) )
8		df-res 4857	. . . 4 |- ( A |` dom ( A |` B ) ) = ( A i^i ( dom ( A |` B ) X. _V ) )
9		dmres 5136	. . . . . . 7 |- dom ( A |` B ) = ( B i^i dom A )
10	9	xpeq1i 4865	. . . . . 6 |- ( dom ( A |` B ) X. _V ) = ( ( B i^i dom A ) X. _V )
11		xpindir 4980	. . . . . 6 |- ( ( B i^i dom A ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
12	10, 11	eqtri 2449	. . . . 5 |- ( dom ( A |` B ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
13	12	ineq2i 3658	. . . 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 2449	. . 3 |- ( A |` dom ( A |` B ) ) = ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) )
15		df-res 4857	. . 3 |- ( `' `' A |` B ) = ( `' `' A i^i ( B X. _V ) )
16	7, 14, 15	3eqtr4i 2459	. 2 |- ( A |` dom ( A |` B ) ) = ( `' `' A |` B )
17		rescnvcnv 5309	. 2 |- ( `' `' A |` B ) = ( A |` B )
18	16, 17	eqtri 2449	1 |- ( A |` dom ( A |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1437   _Vcvv 3078   i^i cin 3432   X. cxp 4843   `'ccnv 4844   dom cdm 4845   |` cres 4847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857

This theorem is referenced by:  imadmres  5338  lindfres  19305  imacmp  20336  metreslem  21301  volres  22376  uhgrares  24907  umgrares  24923  usgrares  24968  uhgres  38491