ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elrnmptg Unicode version

Theorem elrnmptg 4675
Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
elrnmptg  |-  ( A. x  e.  A  B  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)    F( x)    V( x)

Proof of Theorem elrnmptg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rnmpt.1 . . . 4  |-  F  =  ( x  e.  A  |->  B )
21rnmpt 4671 . . 3  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
32eleq2i 2154 . 2  |-  ( C  e.  ran  F  <->  C  e.  { y  |  E. x  e.  A  y  =  B } )
4 r19.29 2506 . . . . 5  |-  ( ( A. x  e.  A  B  e.  V  /\  E. x  e.  A  C  =  B )  ->  E. x  e.  A  ( B  e.  V  /\  C  =  B ) )
5 eleq1 2150 . . . . . . . 8  |-  ( C  =  B  ->  ( C  e.  V  <->  B  e.  V ) )
65biimparc 293 . . . . . . 7  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  V )
7 elex 2630 . . . . . . 7  |-  ( C  e.  V  ->  C  e.  _V )
86, 7syl 14 . . . . . 6  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  _V )
98rexlimivw 2485 . . . . 5  |-  ( E. x  e.  A  ( B  e.  V  /\  C  =  B )  ->  C  e.  _V )
104, 9syl 14 . . . 4  |-  ( ( A. x  e.  A  B  e.  V  /\  E. x  e.  A  C  =  B )  ->  C  e.  _V )
1110ex 113 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  ( E. x  e.  A  C  =  B  ->  C  e. 
_V ) )
12 eqeq1 2094 . . . . 5  |-  ( y  =  C  ->  (
y  =  B  <->  C  =  B ) )
1312rexbidv 2381 . . . 4  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  C  =  B ) )
1413elab3g 2764 . . 3  |-  ( ( E. x  e.  A  C  =  B  ->  C  e.  _V )  -> 
( C  e.  {
y  |  E. x  e.  A  y  =  B }  <->  E. x  e.  A  C  =  B )
)
1511, 14syl 14 . 2  |-  ( A. x  e.  A  B  e.  V  ->  ( C  e.  { y  |  E. x  e.  A  y  =  B }  <->  E. x  e.  A  C  =  B ) )
163, 15syl5bb 190 1  |-  ( A. x  e.  A  B  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619    |-> cmpt 3891   ran crn 4429
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-mpt 3893  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  elrnmpti  4676  fliftel  5554
  Copyright terms: Public domain W3C validator