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Theorem elrn2g 4468
Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
elrn2g (A 𝑉 → (A ran Bxx, A B))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝑉(x)

Proof of Theorem elrn2g
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 opeq2 3541 . . . 4 (y = A → ⟨x, y⟩ = ⟨x, A⟩)
21eleq1d 2103 . . 3 (y = A → (⟨x, y B ↔ ⟨x, A B))
32exbidv 1703 . 2 (y = A → (xx, y Bxx, A B))
4 dfrn3 4467 . 2 ran B = {yxx, y B}
53, 4elab2g 2683 1 (A 𝑉 → (A ran Bxx, A B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wex 1378   wcel 1390  cop 3370  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:  elrng  4469  fvelrn  5241  fo2ndf  5790
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