MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffv4 Unicode version

Theorem dffv4 5688
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 5195), 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 5687 . 2  |-  ( F `
 A )  =  ( iota y y  e.  ( F " { A } ) )
2 df-iota 5381 . 2  |-  ( iota y y  e.  ( F " { A } ) )  = 
U. { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }
3 abid2 2525 . . . . 5  |-  { y  |  y  e.  ( F " { A } ) }  =  ( F " { A } )
43eqeq1i 2415 . . . 4  |-  ( { y  |  y  e.  ( F " { A } ) }  =  { x }  <->  ( F " { A } )  =  { x }
)
54abbii 2520 . . 3  |-  { x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  { x  |  ( F " { A } )  =  { x } }
65unieqi 3989 . 2  |-  U. {
x  |  { y  |  y  e.  ( F " { A } ) }  =  { x } }  =  U. { x  |  ( F " { A } )  =  {
x } }
71, 2, 63eqtri 2432 1  |-  ( F `
 A )  = 
U. { x  |  ( F " { A } )  =  {
x } }
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   {cab 2394   {csn 3778   U.cuni 3979   "cima 4844   iotacio 5379   ` cfv 5417
This theorem is referenced by:  csbfv12gALT  5702  csbfv12gALTVD  28724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-xp 4847  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fv 5425
  Copyright terms: Public domain W3C validator