Theorem resdmres 5498

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 3709	. . . 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 5011	. . . . . 6 |- ( A |` dom A ) = ( A i^i ( dom A X. _V ) )
3		resdm2 5497	. . . . . 6 |- ( A |` dom A ) = `' `' A
4	2, 3	eqtr3i 2498	. . . . 5 |- ( A i^i ( dom A X. _V ) ) = `' `' A
5	4	ineq2i 3697	. . . 4 |- ( ( B X. _V ) i^i ( A i^i ( dom A X. _V ) ) ) = ( ( B X. _V ) i^i `' `' A )
6		incom 3691	. . . 4 |- ( ( B X. _V ) i^i `' `' A ) = ( `' `' A i^i ( B X. _V ) )
7	1, 5, 6	3eqtri 2500	. . 3 |- ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) = ( `' `' A i^i ( B X. _V ) )
8		df-res 5011	. . . 4 |- ( A |` dom ( A |` B ) ) = ( A i^i ( dom ( A |` B ) X. _V ) )
9		dmres 5294	. . . . . . 7 |- dom ( A |` B ) = ( B i^i dom A )
10	9	xpeq1i 5019	. . . . . 6 |- ( dom ( A |` B ) X. _V ) = ( ( B i^i dom A ) X. _V )
11		xpindir 5137	. . . . . 6 |- ( ( B i^i dom A ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
12	10, 11	eqtri 2496	. . . . 5 |- ( dom ( A |` B ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) )
13	12	ineq2i 3697	. . . 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 2496	. . 3 |- ( A |` dom ( A |` B ) ) = ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) )
15		df-res 5011	. . 3 |- ( `' `' A |` B ) = ( `' `' A i^i ( B X. _V ) )
16	7, 14, 15	3eqtr4i 2506	. 2 |- ( A |` dom ( A |` B ) ) = ( `' `' A |` B )
17		rescnvcnv 5470	. 2 |- ( `' `' A |` B ) = ( A |` B )
18	16, 17	eqtri 2496	1 |- ( A |` dom ( A |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1379   _Vcvv 3113   i^i cin 3475   X. cxp 4997   `'ccnv 4998   dom cdm 4999   |` cres 5001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011

This theorem is referenced by:  imadmres  5499  lindfres  18665  imacmp  19703  metreslem  20692  volres  21766  uhgrares  24081  umgrares  24097  usgrares  24142  uhgres  32073