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Theorem elmapssres 7229
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 7226 . . 3  |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
2 fssres 5573 . . 3  |-  ( ( A : C --> B  /\  D  C_  C )  -> 
( A  |`  D ) : D --> B )
31, 2sylan 471 . 2  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( A  |`  D ) : D --> B )
4 elmapex 7225 . . . . 5  |-  ( A  e.  ( B  ^m  C )  ->  ( B  e.  _V  /\  C  e.  _V ) )
54simpld 459 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  B  e.  _V )
65adantr 465 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  ->  B  e.  _V )
74simprd 463 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  C  e.  _V )
8 ssexg 4433 . . . . 5  |-  ( ( D  C_  C  /\  C  e.  _V )  ->  D  e.  _V )
98ancoms 453 . . . 4  |-  ( ( C  e.  _V  /\  D  C_  C )  ->  D  e.  _V )
107, 9sylan 471 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  ->  D  e.  _V )
11 elmapg 7219 . . 3  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( ( A  |`  D )  e.  ( B  ^m  D )  <-> 
( A  |`  D ) : D --> B ) )
126, 10, 11syl2anc 661 . 2  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( ( A  |`  D )  e.  ( B  ^m  D )  <-> 
( A  |`  D ) : D --> B ) )
133, 12mpbird 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    <-> wb 184    /\ wa 369    e. wcel 1756   _Vcvv 2967    C_ wss 3323    |` cres 4837   -->wf 5409  (class class class)co 6086    ^m cmap 7206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-map 7208
This theorem is referenced by:  mdetmul  18409  mapfzcons1cl  29025  mzpcompact2lem  29059  diophin  29082  eldiophss  29084  eldioph4b  29121  nn0gsumfz  30773  lincresunit3lem2  30945
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