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Theorem dffv4g 5286
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 4788), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
dffv4g  |-  ( A  e.  V  ->  ( F `  A )  =  U. { x  |  ( F " { A } )  =  {
x } } )
Distinct variable groups:    x, A    x, F    x, V

Proof of Theorem dffv4g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dffv3g 5285 . 2  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota y
y  e.  ( F
" { A }
) ) )
2 df-iota 4967 . . 3  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }
3 abid2 2208 . . . . . 6  |-  { y  |  y  e.  ( F " { A } ) }  =  ( F " { A } )
43eqeq1i 2095 . . . . 5  |-  ( { y  |  y  e.  ( F " { A } ) }  =  { x }  <->  ( F " { A } )  =  { x }
)
54abbii 2203 . . . 4  |-  { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  { x  |  ( F " { A } )  =  { x } }
65unieqi 3658 . . 3  |-  U. {
x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  U. { x  |  ( F " { A } )  =  {
x } }
72, 6eqtri 2108 . 2  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  ( F " { A } )  =  {
x } }
81, 7syl6eq 2136 1  |-  ( A  e.  V  ->  ( F `  A )  =  U. { x  |  ( F " { A } )  =  {
x } } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   {cab 2074   {csn 3441   U.cuni 3648   "cima 4431   iotacio 4965   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fv 5010
This theorem is referenced by: (None)
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