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Theorem elrn 4693
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 4692 . 2 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
3 df-br 3854 . . 3 (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
43exbii 1542 . 2 (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
52, 4bitr4i 186 1 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104  wex 1427  wcel 1439  Vcvv 2622  cop 3455   class class class wbr 3853  ran crn 4455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-cnv 4462  df-dm 4464  df-rn 4465
This theorem is referenced by:  dmcosseq  4719  rnco  4952  dffo4  5463  rntpos  6038  fclim  10745
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