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Theorem elrn 5336
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 5335 . 2 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
3 df-br 4624 . . 3 (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
43exbii 1771 . 2 (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
52, 4bitr4i 267 1 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1701  wcel 1987  Vcvv 3190  cop 4161   class class class wbr 4623  ran crn 5085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-cnv 5092  df-dm 5094  df-rn 5095
This theorem is referenced by:  dmcosseq  5357  inisegn0  5466  rnco  5610  dffo4  6341  fvclss  6465  rntpos  7325  fpwwe2lem11  9422  fpwwe2lem12  9423  fclim  14234  perfdvf  23607  dftr6  31401  dffr5  31404  brsset  31691  dfon3  31694  brtxpsd  31696  dffix2  31707  elsingles  31720  dfrdg4  31753  undmrnresiss  37430
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