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Theorem elrnmptg 4879
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 4875 . . 3  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
32eleq2i 2244 . 2  |-  ( C  e.  ran  F  <->  C  e.  { y  |  E. x  e.  A  y  =  B } )
4 r19.29 2614 . . . . 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 2240 . . . . . . . 8  |-  ( C  =  B  ->  ( C  e.  V  <->  B  e.  V ) )
65biimparc 299 . . . . . . 7  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  V )
7 elex 2748 . . . . . . 7  |-  ( C  e.  V  ->  C  e.  _V )
86, 7syl 14 . . . . . 6  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  _V )
98rexlimivw 2590 . . . . 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 115 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  ( E. x  e.  A  C  =  B  ->  C  e. 
_V ) )
12 eqeq1 2184 . . . . 5  |-  ( y  =  C  ->  (
y  =  B  <->  C  =  B ) )
1312rexbidv 2478 . . . 4  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  C  =  B ) )
1413elab3g 2888 . . 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, 15bitrid 192 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2737    |-> cmpt 4064   ran crn 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-mpt 4066  df-cnv 4634  df-dm 4636  df-rn 4637
This theorem is referenced by:  elrnmpti  4880  fliftel  5793  2sqlem1  14343
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