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Theorem elunirn2 26086
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 5801 . . . 4  |-  ( B  e.  ( F `  A )  ->  A  e.  dom  F )
2 fveq2 5775 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
32eleq2d 2519 . . . . 5  |-  ( x  =  A  ->  ( B  e.  ( F `  x )  <->  B  e.  ( F `  A ) ) )
43rspcev 3155 . . . 4  |-  ( ( A  e.  dom  F  /\  B  e.  ( F `  A )
)  ->  E. x  e.  dom  F  B  e.  ( F `  x
) )
51, 4mpancom 669 . . 3  |-  ( B  e.  ( F `  A )  ->  E. x  e.  dom  F  B  e.  ( F `  x
) )
65adantl 466 . 2  |-  ( ( Fun  F  /\  B  e.  ( F `  A
) )  ->  E. x  e.  dom  F  B  e.  ( F `  x
) )
7 elunirn 6053 . . 3  |-  ( Fun 
F  ->  ( B  e.  U. ran  F  <->  E. x  e.  dom  F  B  e.  ( F `  x
) ) )
87adantr 465 . 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 369    = wceq 1370    e. wcel 1757   E.wrex 2793   U.cuni 4175   dom cdm 4924   ran crn 4925   Fun wfun 5496   ` cfv 5502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-iota 5465  df-fun 5504  df-fn 5505  df-fv 5510
This theorem is referenced by:  measbasedom  26736  sxbrsigalem0  26806
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