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Theorem elrnmptg 5112
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 5108 . . 3  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
32eleq2i 2499 . 2  |-  ( C  e.  ran  F  <->  C  e.  { y  |  E. x  e.  A  y  =  B } )
4 r19.29 2838 . . . . 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 2495 . . . . . . . 8  |-  ( C  =  B  ->  ( C  e.  V  <->  B  e.  V ) )
65biimparc 474 . . . . . . 7  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  V )
7 elex 2956 . . . . . . 7  |-  ( C  e.  V  ->  C  e.  _V )
86, 7syl 16 . . . . . 6  |-  ( ( B  e.  V  /\  C  =  B )  ->  C  e.  _V )
98rexlimivw 2818 . . . . 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 424 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  ( E. x  e.  A  C  =  B  ->  C  e. 
_V ) )
12 eqeq1 2441 . . . . 5  |-  ( y  =  C  ->  (
y  =  B  <->  C  =  B ) )
1312rexbidv 2718 . . . 4  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  C  =  B ) )
1413elab3g 3080 . . 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 249 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698   _Vcvv 2948    e. cmpt 4258   ran crn 4871
This theorem is referenced by:  elrnmpti  5113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-mpt 4260  df-cnv 4878  df-dm 4880  df-rn 4881
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