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Theorem resdmres 5164
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 3380 . . . 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 4701 . . . . . 6  |-  ( A  |`  dom  A )  =  ( A  i^i  ( dom  A  X.  _V )
)
3 resdm2 5163 . . . . . 6  |-  ( A  |`  dom  A )  =  `' `' A
42, 3eqtr3i 2305 . . . . 5  |-  ( A  i^i  ( dom  A  X.  _V ) )  =  `' `' A
54ineq2i 3367 . . . 4  |-  ( ( B  X.  _V )  i^i  ( A  i^i  ( dom  A  X.  _V )
) )  =  ( ( B  X.  _V )  i^i  `' `' A
)
6 incom 3361 . . . 4  |-  ( ( B  X.  _V )  i^i  `' `' A )  =  ( `' `' A  i^i  ( B  X.  _V ) )
71, 5, 63eqtri 2307 . . 3  |-  ( A  i^i  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )  =  ( `' `' A  i^i  ( B  X.  _V ) )
8 df-res 4701 . . . 4  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  i^i  ( dom  ( A  |`  B )  X.  _V ) )
9 dmres 4976 . . . . . . 7  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
109xpeq1i 4709 . . . . . 6  |-  ( dom  ( A  |`  B )  X.  _V )  =  ( ( B  i^i  dom 
A )  X.  _V )
11 xpindir 4820 . . . . . 6  |-  ( ( B  i^i  dom  A
)  X.  _V )  =  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V ) )
1210, 11eqtri 2303 . . . . 5  |-  ( dom  ( A  |`  B )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V )
)
1312ineq2i 3367 . . . 4  |-  ( A  i^i  ( dom  ( A  |`  B )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )
148, 13eqtri 2303 . . 3  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )
15 df-res 4701 . . 3  |-  ( `' `' A  |`  B )  =  ( `' `' A  i^i  ( B  X.  _V ) )
167, 14, 153eqtr4i 2313 . 2  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( `' `' A  |`  B )
17 rescnvcnv 5135 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
1816, 17eqtri 2303 1  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    i^i cin 3151    X. cxp 4687   `'ccnv 4688   dom cdm 4689    |` cres 4691
This theorem is referenced by:  imadmres  5165  imacmp  17124  metreslem  17926  volres  18887  umgrares  23287  lindfres  26705  usgrares  27514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701
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