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Theorem resfunexgALT 5590
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 5589 but requires ax-pow 4082. (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 4885 . . . 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 4601 . . . 4  |-  ( A
" B )  =  ran  (  A  |`  B )
4 funimaexg 5186 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
53, 4syl5eqelr 2338 . . 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 4707 . 2  |-  ( ( dom  (  A  |`  B )  e.  _V  /\ 
ran  (  A  |`  B )  e.  _V )  ->  ( dom  (  A  |`  B )  X. 
ran  (  A  |`  B ) )  e. 
_V )
8 relres 4890 . . . 4  |-  Rel  ( A  |`  B )
9 relssdmrn 5099 . . . 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 4057 . . 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 2727    C_ wss 3078    X. cxp 4578   dom cdm 4580   ran crn 4581    |` cres 4582   "cima 4583   Rel wrel 4585   Fun wfun 4586
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602
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