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Theorem resfunexgALT 5672
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 5671 but requires ax-pow 4160. (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 4966 . . . 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 4682 . . . 4  |-  ( A
" B )  =  ran  (  A  |`  B )
4 funimaexg 5267 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
53, 4syl5eqelr 2343 . . 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 4788 . 2  |-  ( ( dom  (  A  |`  B )  e.  _V  /\ 
ran  (  A  |`  B )  e.  _V )  ->  ( dom  (  A  |`  B )  X. 
ran  (  A  |`  B ) )  e. 
_V )
8 relres 4971 . . . 4  |-  Rel  ( A  |`  B )
9 relssdmrn 5180 . . . 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 4134 . . 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 654 . 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 1621   _Vcvv 2763    C_ wss 3127    X. cxp 4659   dom cdm 4661   ran crn 4662    |` cres 4663   "cima 4664   Rel wrel 4666   Fun wfun 4667
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683
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