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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
StepHypRef 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
42, 3eqtr3i 2451 . . . . 5  |-  ( A  i^i  ( dom  A  X.  _V ) )  =  `' `' A
54ineq2i 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 ) )
71, 5, 63eqtri 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 )
109xpeq1i 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 ) )
1210, 11eqtri 2449 . . . . 5  |-  ( dom  ( A  |`  B )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V )
)
1312ineq2i 3658 . . . 4  |-  ( A  i^i  ( dom  ( A  |`  B )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )
148, 13eqtri 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 ) )
167, 14, 153eqtr4i 2459 . 2  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( `' `' A  |`  B )
17 rescnvcnv 5309 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
1816, 17eqtri 2449 1  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff setvar class
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
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