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Theorem dffv4g 5384
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 4876), 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 5383 . 2  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota y
y  e.  ( F
" { A }
) ) )
2 df-iota 5056 . . 3  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }
3 abid2 2236 . . . . . 6  |-  { y  |  y  e.  ( F " { A } ) }  =  ( F " { A } )
43eqeq1i 2123 . . . . 5  |-  ( { y  |  y  e.  ( F " { A } ) }  =  { x }  <->  ( F " { A } )  =  { x }
)
54abbii 2231 . . . 4  |-  { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  { x  |  ( F " { A } )  =  { x } }
65unieqi 3714 . . 3  |-  U. {
x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  U. { x  |  ( F " { A } )  =  {
x } }
72, 6eqtri 2136 . 2  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  ( F " { A } )  =  {
x } }
81, 7syl6eq 2164 1  |-  ( A  e.  V  ->  ( F `  A )  =  U. { x  |  ( F " { A } )  =  {
x } } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   {cab 2101   {csn 3495   U.cuni 3704   "cima 4510   iotacio 5054   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fv 5099
This theorem is referenced by: (None)
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