Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elunirn2 Structured version   Unicode version

Theorem elunirn2 27713
Description: Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)
Assertion
Ref Expression
elunirn2  |-  ( ( Fun  F  /\  B  e.  ( F `  A
) )  ->  B  e.  U. ran  F )

Proof of Theorem elunirn2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elfvdm 5874 . . . 4  |-  ( B  e.  ( F `  A )  ->  A  e.  dom  F )
2 fveq2 5848 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
32eleq2d 2524 . . . . 5  |-  ( x  =  A  ->  ( B  e.  ( F `  x )  <->  B  e.  ( F `  A ) ) )
43rspcev 3207 . . . 4  |-  ( ( A  e.  dom  F  /\  B  e.  ( F `  A )
)  ->  E. x  e.  dom  F  B  e.  ( F `  x
) )
51, 4mpancom 667 . . 3  |-  ( B  e.  ( F `  A )  ->  E. x  e.  dom  F  B  e.  ( F `  x
) )
65adantl 464 . 2  |-  ( ( Fun  F  /\  B  e.  ( F `  A
) )  ->  E. x  e.  dom  F  B  e.  ( F `  x
) )
7 elunirn 6138 . . 3  |-  ( Fun 
F  ->  ( B  e.  U. ran  F  <->  E. x  e.  dom  F  B  e.  ( F `  x
) ) )
87adantr 463 . 2  |-  ( ( Fun  F  /\  B  e.  ( F `  A
) )  ->  ( B  e.  U. ran  F  <->  E. x  e.  dom  F  B  e.  ( F `  x ) ) )
96, 8mpbird 232 1  |-  ( ( Fun  F  /\  B  e.  ( F `  A
) )  ->  B  e.  U. ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   U.cuni 4235   dom cdm 4988   ran crn 4989   Fun wfun 5564   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  measbasedom  28413  sxbrsigalem0  28482
  Copyright terms: Public domain W3C validator