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Theorem dffv5 29151
Description: Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
Assertion
Ref Expression
dffv5  |-  ( F `
 A )  = 
U. U. ( { ( F " { A } ) }  i^i  Singletons )

Proof of Theorem dffv5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dffv3 5860 . 2  |-  ( F `
 A )  =  ( iota x x  e.  ( F " { A } ) )
2 dfiota3 29150 . 2  |-  ( iota
x x  e.  ( F " { A } ) )  = 
U. U. ( { {
x  |  x  e.  ( F " { A } ) } }  i^i 
Singletons )
3 abid2 2607 . . . . . 6  |-  { x  |  x  e.  ( F " { A }
) }  =  ( F " { A } )
43sneqi 4038 . . . . 5  |-  { {
x  |  x  e.  ( F " { A } ) } }  =  { ( F " { A } ) }
54ineq1i 3696 . . . 4  |-  ( { { x  |  x  e.  ( F " { A } ) } }  i^i  Singletons )  =  ( { ( F " { A } ) }  i^i  Singletons )
65unieqi 4254 . . 3  |-  U. ( { { x  |  x  e.  ( F " { A } ) } }  i^i  Singletons )  =  U. ( { ( F " { A } ) }  i^i  Singletons )
76unieqi 4254 . 2  |-  U. U. ( { { x  |  x  e.  ( F
" { A }
) } }  i^i  Singletons )  =  U. U. ( { ( F " { A } ) }  i^i  Singletons )
81, 2, 73eqtri 2500 1  |-  ( F `
 A )  = 
U. U. ( { ( F " { A } ) }  i^i  Singletons )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   {cab 2452    i^i cin 3475   {csn 4027   U.cuni 4245   "cima 5002   iotacio 5547   ` cfv 5586   Singletonscsingles 29065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-eprel 4791  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-1st 6781  df-2nd 6782  df-symdif 29045  df-txp 29080  df-singleton 29088  df-singles 29089
This theorem is referenced by:  brapply  29165
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