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Theorem elrnmptg 4761
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 4757 . . 3  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
32eleq2i 2184 . 2  |-  ( C  e.  ran  F  <->  C  e.  { y  |  E. x  e.  A  y  =  B } )
4 r19.29 2546 . . . . 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 2180 . . . . . . . 8  |-  ( C  =  B  ->  ( C  e.  V  <->  B  e.  V ) )
65biimparc 297 . . . . . . 7  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  V )
7 elex 2671 . . . . . . 7  |-  ( C  e.  V  ->  C  e.  _V )
86, 7syl 14 . . . . . 6  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  _V )
98rexlimivw 2522 . . . . 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 114 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  ( E. x  e.  A  C  =  B  ->  C  e. 
_V ) )
12 eqeq1 2124 . . . . 5  |-  ( y  =  C  ->  (
y  =  B  <->  C  =  B ) )
1312rexbidv 2415 . . . 4  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  C  =  B ) )
1413elab3g 2808 . . 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 191 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   _Vcvv 2660    |-> cmpt 3959   ran crn 4510
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-mpt 3961  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  elrnmpti  4762  fliftel  5662
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