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Theorem funexw 6112
Description: Weak version of funex 5739 that holds without ax-coll 4118. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Assertion
Ref Expression
funexw  |-  ( ( Fun  F  /\  dom  F  e.  B  /\  ran  F  e.  C )  ->  F  e.  _V )

Proof of Theorem funexw
StepHypRef Expression
1 xpexg 4740 . . 3  |-  ( ( dom  F  e.  B  /\  ran  F  e.  C
)  ->  ( dom  F  X.  ran  F )  e.  _V )
213adant1 1015 . 2  |-  ( ( Fun  F  /\  dom  F  e.  B  /\  ran  F  e.  C )  -> 
( dom  F  X.  ran  F )  e.  _V )
3 funrel 5233 . . . 4  |-  ( Fun 
F  ->  Rel  F )
4 relssdmrn 5149 . . . 4  |-  ( Rel 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
53, 4syl 14 . . 3  |-  ( Fun 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
653ad2ant1 1018 . 2  |-  ( ( Fun  F  /\  dom  F  e.  B  /\  ran  F  e.  C )  ->  F  C_  ( dom  F  X.  ran  F ) )
72, 6ssexd 4143 1  |-  ( ( Fun  F  /\  dom  F  e.  B  /\  ran  F  e.  C )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 978    e. wcel 2148   _Vcvv 2737    C_ wss 3129    X. cxp 4624   dom cdm 4626   ran crn 4627   Rel wrel 4631   Fun wfun 5210
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-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
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 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637  df-fun 5218
This theorem is referenced by:  mptexw  6113  mpoexw  6213
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