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Theorem resfunexgALT 6112
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 5740 but requires ax-pow 4176 and ax-un 4435. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
resfunexgALT  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )

Proof of Theorem resfunexgALT
StepHypRef Expression
1 dmresexg 4932 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
21adantl 277 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
3 df-ima 4641 . . . 4  |-  ( A
" B )  =  ran  ( A  |`  B )
4 funimaexg 5302 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
53, 4eqeltrrid 2265 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  |`  B )  e.  _V )
62, 5jca 306 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  ( A  |`  B )  e.  _V  /\  ran  ( A  |`  B )  e.  _V ) )
7 xpexg 4742 . 2  |-  ( ( dom  ( A  |`  B )  e.  _V  /\ 
ran  ( A  |`  B )  e.  _V )  ->  ( dom  ( A  |`  B )  X. 
ran  ( A  |`  B ) )  e. 
_V )
8 relres 4937 . . . 4  |-  Rel  ( A  |`  B )
9 relssdmrn 5151 . . . 4  |-  ( Rel  ( A  |`  B )  ->  ( A  |`  B )  C_  ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) ) )
108, 9ax-mp 5 . . 3  |-  ( A  |`  B )  C_  ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) )
11 ssexg 4144 . . 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 424 . 2  |-  ( ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) )  e.  _V  ->  ( A  |`  B )  e.  _V )
136, 7, 123syl 17 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2148   _Vcvv 2739    C_ wss 3131    X. cxp 4626   dom cdm 4628   ran crn 4629    |` cres 4630   "cima 4631   Rel wrel 4633   Fun wfun 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-fun 5220
This theorem is referenced by: (None)
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