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Theorem elrn 4625
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 4624 . 2 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
3 df-br 3806 . . 3 (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
43exbii 1537 . 2 (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
52, 4bitr4i 185 1 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wb 103  wex 1422  wcel 1434  Vcvv 2610  cop 3419   class class class wbr 3805  ran crn 4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by:  dmcosseq  4651  rnco  4877  dffo4  5367  rntpos  5926  fclim  10334
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