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Theorem elmapssres 7436
Description: A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
elmapssres  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( A  |`  D )  e.  ( B  ^m  D ) )

Proof of Theorem elmapssres
StepHypRef Expression
1 elmapi 7433 . . 3  |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
2 fssres 5733 . . 3  |-  ( ( A : C --> B  /\  D  C_  C )  -> 
( A  |`  D ) : D --> B )
31, 2sylan 469 . 2  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( A  |`  D ) : D --> B )
4 elmapex 7432 . . . . 5  |-  ( A  e.  ( B  ^m  C )  ->  ( B  e.  _V  /\  C  e.  _V ) )
54simpld 457 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  B  e.  _V )
65adantr 463 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  ->  B  e.  _V )
74simprd 461 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  C  e.  _V )
8 ssexg 4583 . . . . 5  |-  ( ( D  C_  C  /\  C  e.  _V )  ->  D  e.  _V )
98ancoms 451 . . . 4  |-  ( ( C  e.  _V  /\  D  C_  C )  ->  D  e.  _V )
107, 9sylan 469 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  ->  D  e.  _V )
116, 10elmapd 7426 . 2  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( ( A  |`  D )  e.  ( B  ^m  D )  <-> 
( A  |`  D ) : D --> B ) )
123, 11mpbird 232 1  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( A  |`  D )  e.  ( B  ^m  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   _Vcvv 3106    C_ wss 3461    |` cres 4990   -->wf 5566  (class class class)co 6270    ^m cmap 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414
This theorem is referenced by:  nn0gsumfz  17207  mdetmul  19292  mapfzcons1cl  30890  mzpcompact2lem  30923  diophin  30945  eldiophss  30947  eldioph4b  30984  mccllem  31844  lincresunit3lem2  33335
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