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Theorem resfunexgALT 5974
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 5607 but requires ax-pow 4066 and ax-un 4323. (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 4810 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
21adantl 273 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
3 df-ima 4520 . . . 4  |-  ( A
" B )  =  ran  ( A  |`  B )
4 funimaexg 5175 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
53, 4eqeltrrid 2203 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  |`  B )  e.  _V )
62, 5jca 302 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  ( A  |`  B )  e.  _V  /\  ran  ( A  |`  B )  e.  _V ) )
7 xpexg 4621 . 2  |-  ( ( dom  ( A  |`  B )  e.  _V  /\ 
ran  ( A  |`  B )  e.  _V )  ->  ( dom  ( A  |`  B )  X. 
ran  ( A  |`  B ) )  e. 
_V )
8 relres 4815 . . . 4  |-  Rel  ( A  |`  B )
9 relssdmrn 5027 . . . 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 4035 . . 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 418 . 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 103    e. wcel 1463   _Vcvv 2658    C_ wss 3039    X. cxp 4505   dom cdm 4507   ran crn 4508    |` cres 4509   "cima 4510   Rel wrel 4512   Fun wfun 5085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-fun 5093
This theorem is referenced by: (None)
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