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Theorem elmapssres 26298
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 6935 . . 3  |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
2 fssres 5514 . . 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 6934 . . . . 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 4262 . . . . 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 6928 . . 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 1715   _Vcvv 2873    C_ wss 3238    |` cres 4794   -->wf 5354  (class class class)co 5981    ^m cmap 6915
This theorem is referenced by:  mapfzcons1cl  26301  mzpcompact2lem  26335  diophin  26358  eldiophss  26360  eldioph4b  26400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-map 6917
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