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Theorem elrnmpt1s 5239
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
Hypotheses
Ref Expression
rnmpt.1  |-  F  =  ( x  e.  A  |->  B )
elrnmpt1s.1  |-  ( x  =  D  ->  B  =  C )
Assertion
Ref Expression
elrnmpt1s  |-  ( ( D  e.  A  /\  C  e.  V )  ->  C  e.  ran  F
)
Distinct variable groups:    x, C    x, A    x, D
Allowed substitution hints:    B( x)    F( x)    V( x)

Proof of Theorem elrnmpt1s
StepHypRef Expression
1 eqid 2454 . . 3  |-  C  =  C
2 elrnmpt1s.1 . . . . 5  |-  ( x  =  D  ->  B  =  C )
32eqeq2d 2468 . . . 4  |-  ( x  =  D  ->  ( C  =  B  <->  C  =  C ) )
43rspcev 3207 . . 3  |-  ( ( D  e.  A  /\  C  =  C )  ->  E. x  e.  A  C  =  B )
51, 4mpan2 669 . 2  |-  ( D  e.  A  ->  E. x  e.  A  C  =  B )
6 rnmpt.1 . . . 4  |-  F  =  ( x  e.  A  |->  B )
76elrnmpt 5238 . . 3  |-  ( C  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
87biimparc 485 . 2  |-  ( ( E. x  e.  A  C  =  B  /\  C  e.  V )  ->  C  e.  ran  F
)
95, 8sylan 469 1  |-  ( ( D  e.  A  /\  C  e.  V )  ->  C  e.  ran  F
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805    |-> cmpt 4497   ran crn 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-mpt 4499  df-cnv 4996  df-dm 4998  df-rn 4999
This theorem is referenced by:  wunex2  9105  dfod2  16785  dprd2dlem1  17285  dprd2da  17286  ordtbaslem  19856  subgntr  20771  opnsubg  20772  tgpconcomp  20777  tsmsxplem1  20821  xrge0gsumle  21504  xrge0tsms  21505  minveclem3b  22009  minveclem3  22010  minveclem4  22013  efsubm  23104  dchrisum0fno1  23894  xrge0tsmsd  28010  esumcvg  28315  esum2d  28322  msubco  29155
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