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Theorem dffv4g 5206
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 4724), 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 5205 . 2  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota y
y  e.  ( F
" { A }
) ) )
2 df-iota 4897 . . 3  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }
3 abid2 2200 . . . . . 6  |-  { y  |  y  e.  ( F " { A } ) }  =  ( F " { A } )
43eqeq1i 2089 . . . . 5  |-  ( { y  |  y  e.  ( F " { A } ) }  =  { x }  <->  ( F " { A } )  =  { x }
)
54abbii 2195 . . . 4  |-  { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  { x  |  ( F " { A } )  =  { x } }
65unieqi 3619 . . 3  |-  U. {
x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  U. { x  |  ( F " { A } )  =  {
x } }
72, 6eqtri 2102 . 2  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  ( F " { A } )  =  {
x } }
81, 7syl6eq 2130 1  |-  ( A  e.  V  ->  ( F `  A )  =  U. { x  |  ( F " { A } )  =  {
x } } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   {cab 2068   {csn 3406   U.cuni 3609   "cima 4374   iotacio 4895   ` cfv 4932
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fv 4940
This theorem is referenced by: (None)
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