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Theorem elrnmpt1s 5188
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 2451 . . 3  |-  C  =  C
2 elrnmpt1s.1 . . . . 5  |-  ( x  =  D  ->  B  =  C )
32eqeq2d 2465 . . . 4  |-  ( x  =  D  ->  ( C  =  B  <->  C  =  C ) )
43rspcev 3172 . . 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 5187 . . 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 1370    e. wcel 1758   E.wrex 2796    |-> cmpt 4451   ran crn 4942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-mpt 4453  df-cnv 4949  df-dm 4951  df-rn 4952
This theorem is referenced by:  wunex2  9009  dfod2  16178  dprd2dlem1  16654  dprd2da  16655  ordtbaslem  18917  subgntr  19802  opnsubg  19803  tgpconcomp  19808  tsmsxplem1  19852  xrge0gsumle  20535  xrge0tsms  20536  minveclem3b  21040  minveclem3  21041  minveclem4  21044  dchrisum0fno1  22886  xrge0tsmsd  26391  esumcvg  26673
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