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Theorem resfunexgALT 5699
Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5698 but requires ax-pow 4187. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.)
Assertion
Ref Expression
resfunexgALT  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )

Proof of Theorem resfunexgALT
StepHypRef Expression
1 dmresexg 4977 . . . 4  |-  ( B  e.  C  ->  dom  (  A  |`  B )  e.  _V )
21adantl 454 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  (  A  |`  B )  e.  _V )
3 df-ima 4701 . . . 4  |-  ( A
" B )  =  ran  (  A  |`  B )
4 funimaexg 5294 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
53, 4syl5eqelr 2369 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  (  A  |`  B )  e.  _V )
62, 5jca 520 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  (  A  |`  B )  e.  _V  /\  ran  (  A  |`  B )  e.  _V ) )
7 xpexg 4799 . 2  |-  ( ( dom  (  A  |`  B )  e.  _V  /\ 
ran  (  A  |`  B )  e.  _V )  ->  ( dom  (  A  |`  B )  X. 
ran  (  A  |`  B ) )  e. 
_V )
8 relres 4982 . . . 4  |-  Rel  ( A  |`  B )
9 relssdmrn 5191 . . . 4  |-  ( Rel  ( A  |`  B )  ->  ( A  |`  B )  C_  ( dom  (  A  |`  B )  X.  ran  (  A  |`  B ) ) )
108, 9ax-mp 10 . . 3  |-  ( A  |`  B )  C_  ( dom  (  A  |`  B )  X.  ran  (  A  |`  B ) )
11 ssexg 4161 . . 3  |-  ( ( ( A  |`  B ) 
C_  ( dom  (  A  |`  B )  X. 
ran  (  A  |`  B ) )  /\  ( dom  (  A  |`  B )  X.  ran  (  A  |`  B ) )  e.  _V )  ->  ( A  |`  B )  e.  _V )
1210, 11mpan 653 . 2  |-  ( ( dom  (  A  |`  B )  X.  ran  (  A  |`  B ) )  e.  _V  ->  ( A  |`  B )  e.  _V )
136, 7, 123syl 20 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1685   _Vcvv 2789    C_ wss 3153    X. cxp 4686   dom cdm 4688   ran crn 4689    |` cres 4690   "cima 4691   Rel wrel 4693   Fun wfun 5215
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223
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