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Theorem elrnmpt1s 5082
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 3068 . . 3  |-  ( ( D  e.  A  /\  C  =  C )  ->  E. x  e.  A  C  =  B )
51, 4mpan2 671 . 2  |-  ( D  e.  A  ->  E. x  e.  A  C  =  B )
6 rnmpt.1 . . . 4  |-  F  =  ( x  e.  A  |->  B )
76elrnmpt 5081 . . 3  |-  ( C  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
87biimparc 487 . 2  |-  ( ( E. x  e.  A  C  =  B  /\  C  e.  V )  ->  C  e.  ran  F
)
95, 8sylan 471 1  |-  ( ( D  e.  A  /\  C  e.  V )  ->  C  e.  ran  F
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2711    e. cmpt 4345   ran crn 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-mpt 4347  df-cnv 4843  df-dm 4845  df-rn 4846
This theorem is referenced by:  wunex2  8897  dfod2  16056  dprd2dlem1  16528  dprd2da  16529  ordtbaslem  18767  subgntr  19652  opnsubg  19653  tgpconcomp  19658  tsmsxplem1  19702  xrge0gsumle  20385  xrge0tsms  20386  minveclem3b  20890  minveclem3  20891  minveclem4  20894  dchrisum0fno1  22735  xrge0tsmsd  26204  esumcvg  26487
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