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Theorem elmapssres 26809
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 7067 . . 3  |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
2 fssres 5639 . . 3  |-  ( ( A : C --> B  /\  D  C_  C )  -> 
( A  |`  D ) : D --> B )
31, 2sylan 459 . 2  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( A  |`  D ) : D --> B )
4 elmapex 7066 . . . . 5  |-  ( A  e.  ( B  ^m  C )  ->  ( B  e.  _V  /\  C  e.  _V ) )
54simpld 447 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  B  e.  _V )
65adantr 453 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  ->  B  e.  _V )
74simprd 451 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  C  e.  _V )
8 ssexg 4378 . . . . 5  |-  ( ( D  C_  C  /\  C  e.  _V )  ->  D  e.  _V )
98ancoms 441 . . . 4  |-  ( ( C  e.  _V  /\  D  C_  C )  ->  D  e.  _V )
107, 9sylan 459 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  ->  D  e.  _V )
11 elmapg 7060 . . 3  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( ( A  |`  D )  e.  ( B  ^m  D )  <-> 
( A  |`  D ) : D --> B ) )
126, 10, 11syl2anc 644 . 2  |-  ( ( A  e.  ( B  ^m  C )  /\  D  C_  C )  -> 
( ( A  |`  D )  e.  ( B  ^m  D )  <-> 
( A  |`  D ) : D --> B ) )
133, 12mpbird 225 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 178    /\ wa 360    e. wcel 1727   _Vcvv 2962    C_ wss 3306    |` cres 4909   -->wf 5479  (class class class)co 6110    ^m cmap 7047
This theorem is referenced by:  mapfzcons1cl  26812  mzpcompact2lem  26846  diophin  26869  eldiophss  26871  eldioph4b  26910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-map 7049
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