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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
StepHypRef 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
42, 3eqtr3i 2498 . . . . 5  |-  ( A  i^i  ( dom  A  X.  _V ) )  =  `' `' A
54ineq2i 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 ) )
71, 5, 63eqtri 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 )
109xpeq1i 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 ) )
1210, 11eqtri 2496 . . . . 5  |-  ( dom  ( A  |`  B )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( dom  A  X.  _V )
)
1312ineq2i 3697 . . . 4  |-  ( A  i^i  ( dom  ( A  |`  B )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( dom  A  X.  _V ) ) )
148, 13eqtri 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 ) )
167, 14, 153eqtr4i 2506 . 2  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( `' `' A  |`  B )
17 rescnvcnv 5470 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
1816, 17eqtri 2496 1  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff setvar class
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
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