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Theorem residcp 25077
Description: The intersection of the identity function with a square cross product. (Contributed by FL, 2-Aug-2009.)
Assertion
Ref Expression
residcp  |-  (  _I 
i^i  ( A  X.  A ) )  =  (  _I  |`  A )

Proof of Theorem residcp
StepHypRef Expression
1 ssid 3197 . . . . 5  |-  A  C_  A
2 ssv 3198 . . . . 5  |-  A  C_  _V
3 xpss12 4792 . . . . 5  |-  ( ( A  C_  A  /\  A  C_  _V )  -> 
( A  X.  A
)  C_  ( A  X.  _V ) )
41, 2, 3mp2an 653 . . . 4  |-  ( A  X.  A )  C_  ( A  X.  _V )
5 sslin 3395 . . . 4  |-  ( ( A  X.  A ) 
C_  ( A  X.  _V )  ->  (  _I 
i^i  ( A  X.  A ) )  C_  (  _I  i^i  ( A  X.  _V ) ) )
64, 5ax-mp 8 . . 3  |-  (  _I 
i^i  ( A  X.  A ) )  C_  (  _I  i^i  ( A  X.  _V ) )
7 df-res 4701 . . 3  |-  (  _I  |`  A )  =  (  _I  i^i  ( A  X.  _V ) )
86, 7sseqtr4i 3211 . 2  |-  (  _I 
i^i  ( A  X.  A ) )  C_  (  _I  |`  A )
9 resss 4979 . . 3  |-  (  _I  |`  A )  C_  _I
10 scprefat2 25072 . . 3  |-  (  _I  |`  A )  C_  ( A  X.  A )
119, 10ssini 3392 . 2  |-  (  _I  |`  A )  C_  (  _I  i^i  ( A  X.  A ) )
128, 11eqssi 3195 1  |-  (  _I 
i^i  ( A  X.  A ) )  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    i^i cin 3151    C_ wss 3152    _I cid 4304    X. cxp 4687    |` cres 4691
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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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