Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfaimafn Structured version   Unicode version

Theorem dfaimafn 30195
Description: Alternate definition of the image of a function, analogous to dfimafn 5825. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
dfaimafn  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F''' x )  =  y } )
Distinct variable groups:    x, y, A    x, F, y

Proof of Theorem dfaimafn
StepHypRef Expression
1 ssel 3434 . . . . . 6  |-  ( A 
C_  dom  F  ->  ( x  e.  A  ->  x  e.  dom  F ) )
2 funbrafvb 30186 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F''' x )  =  y  <->  x F
y ) )
32ex 434 . . . . . 6  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  (
( F''' x )  =  y  <-> 
x F y ) ) )
41, 3syl9r 72 . . . . 5  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( x  e.  A  ->  (
( F''' x )  =  y  <-> 
x F y ) ) ) )
54imp31 432 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( ( F''' x )  =  y  <-> 
x F y ) )
65rexbidva 2812 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. x  e.  A  ( F''' x )  =  y  <->  E. x  e.  A  x F
y ) )
76abbidv 2584 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  ->  { y  |  E. x  e.  A  ( F''' x )  =  y }  =  { y  |  E. x  e.  A  x F y } )
8 dfima2 5255 . 2  |-  ( F
" A )  =  { y  |  E. x  e.  A  x F y }
97, 8syl6reqr 2509 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F''' x )  =  y } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1757   {cab 2435   E.wrex 2793    C_ wss 3412   class class class wbr 4376   dom cdm 4924   "cima 4927   Fun wfun 5496  '''cafv 30142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-fv 5510  df-dfat 30144  df-afv 30145
This theorem is referenced by:  dfaimafn2  30196
  Copyright terms: Public domain W3C validator