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Theorem elrnmpt1s 5120
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 2438 . . 3  |-  C  =  C
2 elrnmpt1s.1 . . . . 5  |-  ( x  =  D  ->  B  =  C )
32eqeq2d 2449 . . . 4  |-  ( x  =  D  ->  ( C  =  B  <->  C  =  C ) )
43rspcev 3054 . . 3  |-  ( ( D  e.  A  /\  C  =  C )  ->  E. x  e.  A  C  =  B )
51, 4mpan2 654 . 2  |-  ( D  e.  A  ->  E. x  e.  A  C  =  B )
6 rnmpt.1 . . . 4  |-  F  =  ( x  e.  A  |->  B )
76elrnmpt 5119 . . 3  |-  ( C  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
87biimparc 475 . 2  |-  ( ( E. x  e.  A  C  =  B  /\  C  e.  V )  ->  C  e.  ran  F
)
95, 8sylan 459 1  |-  ( ( D  e.  A  /\  C  e.  V )  ->  C  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    e. cmpt 4268   ran crn 4881
This theorem is referenced by:  wunex2  8615  dfod2  15202  dprd2dlem1  15601  dprd2da  15602  ordtbaslem  17254  subgntr  18138  opnsubg  18139  tgpconcomp  18144  tsmsxplem1  18184  xrge0gsumle  18866  xrge0tsms  18867  minveclem3b  19331  minveclem3  19332  minveclem4  19335  dchrisum0fno1  21207  xrge0tsmsd  24225  esumcvg  24478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891
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