ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  relrnfvex Unicode version

Theorem relrnfvex 5514
Description: If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)
Assertion
Ref Expression
relrnfvex  |-  ( ( Rel  F  /\  ran  F  e.  _V )  -> 
( F `  A
)  e.  _V )

Proof of Theorem relrnfvex
StepHypRef Expression
1 relfvssunirn 5512 . 2  |-  ( Rel 
F  ->  ( F `  A )  C_  U. ran  F )
2 uniexg 4424 . 2  |-  ( ran 
F  e.  _V  ->  U.
ran  F  e.  _V )
3 ssexg 4128 . 2  |-  ( ( ( F `  A
)  C_  U. ran  F  /\  U. ran  F  e. 
_V )  ->  ( F `  A )  e.  _V )
41, 2, 3syl2an 287 1  |-  ( ( Rel  F  /\  ran  F  e.  _V )  -> 
( F `  A
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2141   _Vcvv 2730    C_ wss 3121   U.cuni 3796   ran crn 4612   Rel wrel 4616   ` cfv 5198
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-iota 5160  df-fv 5206
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator