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Theorem afvelrnb 27994
 Description: A member of a function's range is a value of the function, analogous to fvelrnb 5766 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvelrnb '''
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem afvelrnb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fnrnafv 27993 . . . 4 '''
21adantr 452 . . 3 '''
32eleq2d 2502 . 2 '''
4 eqeq1 2441 . . . . . 6 ''' '''
5 eqcom 2437 . . . . . 6 ''' '''
64, 5syl6bb 253 . . . . 5 ''' '''
76rexbidv 2718 . . . 4 ''' '''
87elabg 3075 . . 3 ''' '''
98adantl 453 . 2 ''' '''
103, 9bitrd 245 1 '''
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cab 2421  wrex 2698   crn 4871   wfn 5441  '''cafv 27939 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-dfat 27941  df-afv 27942
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