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Theorem elmapssres 26792
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 6792 . . 3  |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
2 fssres 5408 . . 3  |-  ( ( A : C --> B  /\  D  C_  C )  -> 
( A  |`  D ) : D --> B )
31, 2sylan 457 . 2  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( A  |`  D ) : D --> B )
4 elmapex 6791 . . . . 5  |-  ( A  e.  ( B  ^m  C )  ->  ( B  e.  _V  /\  C  e.  _V ) )
54simpld 445 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  B  e.  _V )
65adantr 451 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  ->  B  e.  _V )
74simprd 449 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  C  e.  _V )
8 ssexg 4160 . . . . 5  |-  ( ( D  C_  C  /\  C  e.  _V )  ->  D  e.  _V )
98ancoms 439 . . . 4  |-  ( ( C  e.  _V  /\  D  C_  C )  ->  D  e.  _V )
107, 9sylan 457 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  ->  D  e.  _V )
11 elmapg 6785 . . 3  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( ( A  |`  D )  e.  ( B  ^m  D )  <-> 
( A  |`  D ) : D --> B ) )
126, 10, 11syl2anc 642 . 2  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( ( A  |`  D )  e.  ( B  ^m  D )  <-> 
( A  |`  D ) : D --> B ) )
133, 12mpbird 223 1  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( A  |`  D )  e.  ( B  ^m  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   _Vcvv 2788    C_ wss 3152    |` cres 4691   -->wf 5251  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  mapfzcons1cl  26795  mzpcompact2lem  26829  diophin  26852  eldiophss  26854  eldioph4b  26894
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774
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