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Theorem elrn 4520
 Description: Membership in a range. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
elrn.1 A V
Assertion
Ref Expression
elrn (A ran Bx xBA)
Distinct variable groups:   x,A   x,B

Proof of Theorem elrn
StepHypRef Expression
1 elrn.1 . . 3 A V
21elrn2 4519 . 2 (A ran Bxx, A B)
3 df-br 3756 . . 3 (xBA ↔ ⟨x, A B)
43exbii 1493 . 2 (x xBAxx, A B)
52, 4bitr4i 176 1 (A ran Bx xBA)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   class class class wbr 3755  ran crn 4289 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299 This theorem is referenced by:  dmcosseq  4546  rnco  4770  dffo4  5258  rntpos  5813
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