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Theorem elrnmptg 5252
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 5248 . . 3  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
32eleq2i 2545 . 2  |-  ( C  e.  ran  F  <->  C  e.  { y  |  E. x  e.  A  y  =  B } )
4 r19.29 2997 . . . . 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 2539 . . . . . . . 8  |-  ( C  =  B  ->  ( C  e.  V  <->  B  e.  V ) )
65biimparc 487 . . . . . . 7  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  V )
7 elex 3122 . . . . . . 7  |-  ( C  e.  V  ->  C  e.  _V )
86, 7syl 16 . . . . . 6  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  _V )
98rexlimivw 2952 . . . . 5  |-  ( E. x  e.  A  ( B  e.  V  /\  C  =  B )  ->  C  e.  _V )
104, 9syl 16 . . . 4  |-  ( ( A. x  e.  A  B  e.  V  /\  E. x  e.  A  C  =  B )  ->  C  e.  _V )
1110ex 434 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  ( E. x  e.  A  C  =  B  ->  C  e. 
_V ) )
12 eqeq1 2471 . . . . 5  |-  ( y  =  C  ->  (
y  =  B  <->  C  =  B ) )
1312rexbidv 2973 . . . 4  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  C  =  B ) )
1413elab3g 3256 . . 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 16 . 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 257 1  |-  ( A. x  e.  A  B  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113    |-> cmpt 4505   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  elrnmpti  5253
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