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Theorem dffv4 5522
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 5041), 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 5521 . 2  |-  ( F `
 A )  =  ( iota y y  e.  ( F " { A } ) )
2 df-iota 5219 . 2  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }
3 abid2 2400 . . . . 5  |-  { y  |  y  e.  ( F " { A } ) }  =  ( F " { A } )
43eqeq1i 2290 . . . 4  |-  ( { y  |  y  e.  ( F " { A } ) }  =  { x }  <->  ( F " { A } )  =  { x }
)
54abbii 2395 . . 3  |-  { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  { x  |  ( F " { A } )  =  { x } }
65unieqi 3837 . 2  |-  U. {
x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  U. { x  |  ( F " { A } )  =  {
x } }
71, 2, 63eqtri 2307 1  |-  ( F `
 A )  = 
U. { x  |  ( F " { A } )  =  {
x } }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   {csn 3640   U.cuni 3827   "cima 4692   iotacio 5217   ` cfv 5255
This theorem is referenced by:  csbfv12gALT  5536  dfafv2  27995  csbfv12gALTVD  28675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263
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