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Theorem dffv4 5629
Description: The previous definition of function value, from before the 
iota operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 5144), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
dffv4  |-  ( F `
 A )  = 
U. { x  |  ( F " { A } )  =  {
x } }
Distinct variable groups:    x, A    x, F

Proof of Theorem dffv4
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dffv3 5628 . 2  |-  ( F `
 A )  =  ( iota y y  e.  ( F " { A } ) )
2 df-iota 5322 . 2  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }
3 abid2 2483 . . . . 5  |-  { y  |  y  e.  ( F " { A } ) }  =  ( F " { A } )
43eqeq1i 2373 . . . 4  |-  ( { y  |  y  e.  ( F " { A } ) }  =  { x }  <->  ( F " { A } )  =  { x }
)
54abbii 2478 . . 3  |-  { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  { x  |  ( F " { A } )  =  { x } }
65unieqi 3939 . 2  |-  U. {
x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  U. { x  |  ( F " { A } )  =  {
x } }
71, 2, 63eqtri 2390 1  |-  ( F `
 A )  = 
U. { x  |  ( F " { A } )  =  {
x } }
Colors of variables: wff set class
Syntax hints:    = wceq 1647    e. wcel 1715   {cab 2352   {csn 3729   U.cuni 3929   "cima 4795   iotacio 5320   ` cfv 5358
This theorem is referenced by:  csbfv12gALT  5643  csbfv12gALTVD  28439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-xp 4798  df-cnv 4800  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fv 5366
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