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Theorem elrn 5270
Description: Membership in a range. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
elrn.1 𝐴 ∈ V
Assertion
Ref Expression
elrn (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elrn
StepHypRef Expression
1 elrn.1 . . 3 𝐴 ∈ V
21elrn2 5269 . 2 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
3 df-br 4574 . . 3 (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
43exbii 1762 . 2 (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
52, 4bitr4i 265 1 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wex 1694  wcel 1975  Vcvv 3168  cop 4126   class class class wbr 4573  ran crn 5025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-cnv 5032  df-dm 5034  df-rn 5035
This theorem is referenced by:  dmcosseq  5291  inisegn0  5399  rnco  5540  dffo4  6264  fvclss  6378  rntpos  7225  fpwwe2lem11  9314  fpwwe2lem12  9315  fclim  14074  perfdvf  23386  dftr6  30695  dffr5  30698  brsset  30968  dfon3  30971  brtxpsd  30973  dffix2  30984  elsingles  30997  dfrdg4  31030  undmrnresiss  36728
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