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Theorem dffv4g 5511
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 4996), 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 5510 . 2  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota y
y  e.  ( F
" { A }
) ) )
2 df-iota 5177 . . 3  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }
3 abid2 2298 . . . . . 6  |-  { y  |  y  e.  ( F " { A } ) }  =  ( F " { A } )
43eqeq1i 2185 . . . . 5  |-  ( { y  |  y  e.  ( F " { A } ) }  =  { x }  <->  ( F " { A } )  =  { x }
)
54abbii 2293 . . . 4  |-  { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  { x  |  ( F " { A } )  =  { x } }
65unieqi 3819 . . 3  |-  U. {
x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  U. { x  |  ( F " { A } )  =  {
x } }
72, 6eqtri 2198 . 2  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  ( F " { A } )  =  {
x } }
81, 7eqtrdi 2226 1  |-  ( A  e.  V  ->  ( F `  A )  =  U. { x  |  ( F " { A } )  =  {
x } } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   {cab 2163   {csn 3592   U.cuni 3809   "cima 4628   iotacio 5175   ` cfv 5215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-xp 4631  df-cnv 4633  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fv 5223
This theorem is referenced by: (None)
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