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Theorem resdmres 5333
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
StepHypRef Expression
1 in12 3634 . . . 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 4851 . . . . . 6  |-  ( A  |`  dom  A )  =  ( A  i^i  ( dom  A  X.  _V )
)
3 resdm2 5332 . . . . . 6  |-  ( A  |`  dom  A )  =  `' `' A
42, 3eqtr3i 2495 . . . . 5  |-  ( A  i^i  ( dom  A  X.  _V ) )  =  `' `' A
54ineq2i 3622 . . . 4  |-  ( ( B  X.  _V )  i^i  ( A  i^i  ( dom  A  X.  _V )
) )  =  ( ( B  X.  _V )  i^i  `' `' A
)
6 incom 3616 . . . 4  |-  ( ( B  X.  _V )  i^i  `' `' A )  =  ( `' `' A  i^i  ( B  X.  _V ) )
71, 5, 63eqtri 2497 . . 3  |-  ( A  i^i  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )  =  ( `' `' A  i^i  ( B  X.  _V ) )
8 df-res 4851 . . . 4  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  i^i  ( dom  ( A  |`  B )  X.  _V ) )
9 dmres 5131 . . . . . . 7  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
109xpeq1i 4859 . . . . . 6  |-  ( dom  ( A  |`  B )  X.  _V )  =  ( ( B  i^i  dom 
A )  X.  _V )
11 xpindir 4974 . . . . . 6  |-  ( ( B  i^i  dom  A
)  X.  _V )  =  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V ) )
1210, 11eqtri 2493 . . . . 5  |-  ( dom  ( A  |`  B )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V )
)
1312ineq2i 3622 . . . 4  |-  ( A  i^i  ( dom  ( A  |`  B )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )
148, 13eqtri 2493 . . 3  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )
15 df-res 4851 . . 3  |-  ( `' `' A  |`  B )  =  ( `' `' A  i^i  ( B  X.  _V ) )
167, 14, 153eqtr4i 2503 . 2  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( `' `' A  |`  B )
17 rescnvcnv 5305 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
1816, 17eqtri 2493 1  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   _Vcvv 3031    i^i cin 3389    X. cxp 4837   `'ccnv 4838   dom cdm 4839    |` cres 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851
This theorem is referenced by:  imadmres  5334  lindfres  19458  imacmp  20489  metreslem  21455  volres  22560  uhgrares  25114  umgrares  25130  usgrares  25175  resresdm  39154  uhgres  40199
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