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Theorem dffv5 30249
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 5844 . 2  |-  ( F `
 A )  =  ( iota x x  e.  ( F " { A } ) )
2 dfiota3 30248 . 2  |-  ( iota
x x  e.  ( F " { A } ) )  = 
U. U. ( { {
x  |  x  e.  ( F " { A } ) } }  i^i 
Singletons )
3 abid2 2542 . . . . . 6  |-  { x  |  x  e.  ( F " { A }
) }  =  ( F " { A } )
43sneqi 3982 . . . . 5  |-  { {
x  |  x  e.  ( F " { A } ) } }  =  { ( F " { A } ) }
54ineq1i 3636 . . . 4  |-  ( { { x  |  x  e.  ( F " { A } ) } }  i^i  Singletons )  =  ( { ( F " { A } ) }  i^i  Singletons )
65unieqi 4199 . . 3  |-  U. ( { { x  |  x  e.  ( F " { A } ) } }  i^i  Singletons )  =  U. ( { ( F " { A } ) }  i^i  Singletons )
76unieqi 4199 . 2  |-  U. U. ( { { x  |  x  e.  ( F
" { A }
) } }  i^i  Singletons )  =  U. U. ( { ( F " { A } ) }  i^i  Singletons )
81, 2, 73eqtri 2435 1  |-  ( F `
 A )  = 
U. U. ( { ( F " { A } ) }  i^i  Singletons )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   {cab 2387    i^i cin 3412   {csn 3971   U.cuni 4190   "cima 4825   iotacio 5530   ` cfv 5568   Singletonscsingles 30163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-symdif 3669  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-eprel 4733  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-1st 6783  df-2nd 6784  df-txp 30178  df-singleton 30186  df-singles 30187
This theorem is referenced by:  brapply  30263
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