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Theorem resdmres 5364
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 3554 . . . 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 4893 . . . . . 6  |-  ( A  |`  dom  A )  =  ( A  i^i  ( dom  A  X.  _V )
)
3 resdm2 5363 . . . . . 6  |-  ( A  |`  dom  A )  =  `' `' A
42, 3eqtr3i 2460 . . . . 5  |-  ( A  i^i  ( dom  A  X.  _V ) )  =  `' `' A
54ineq2i 3541 . . . 4  |-  ( ( B  X.  _V )  i^i  ( A  i^i  ( dom  A  X.  _V )
) )  =  ( ( B  X.  _V )  i^i  `' `' A
)
6 incom 3535 . . . 4  |-  ( ( B  X.  _V )  i^i  `' `' A )  =  ( `' `' A  i^i  ( B  X.  _V ) )
71, 5, 63eqtri 2462 . . 3  |-  ( A  i^i  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )  =  ( `' `' A  i^i  ( B  X.  _V ) )
8 df-res 4893 . . . 4  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  i^i  ( dom  ( A  |`  B )  X.  _V ) )
9 dmres 5170 . . . . . . 7  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
109xpeq1i 4901 . . . . . 6  |-  ( dom  ( A  |`  B )  X.  _V )  =  ( ( B  i^i  dom 
A )  X.  _V )
11 xpindir 5012 . . . . . 6  |-  ( ( B  i^i  dom  A
)  X.  _V )  =  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V ) )
1210, 11eqtri 2458 . . . . 5  |-  ( dom  ( A  |`  B )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V )
)
1312ineq2i 3541 . . . 4  |-  ( A  i^i  ( dom  ( A  |`  B )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )
148, 13eqtri 2458 . . 3  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )
15 df-res 4893 . . 3  |-  ( `' `' A  |`  B )  =  ( `' `' A  i^i  ( B  X.  _V ) )
167, 14, 153eqtr4i 2468 . 2  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( `' `' A  |`  B )
17 rescnvcnv 5335 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
1816, 17eqtri 2458 1  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   _Vcvv 2958    i^i cin 3321    X. cxp 4879   `'ccnv 4880   dom cdm 4881    |` cres 4883
This theorem is referenced by:  imadmres  5365  imacmp  17465  metreslem  18397  volres  19429  uhgrares  21348  umgrares  21364  usgrares  21394  lindfres  27284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893
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