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Theorem elrn2g 4729
Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
elrn2g (𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 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 3706 . . . 4 (𝑦 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐴⟩)
21eleq1d 2208 . . 3 (𝑦 = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
32exbidv 1797 . 2 (𝑦 = 𝐴 → (∃𝑥𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
4 dfrn3 4728 . 2 ran 𝐵 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐵}
53, 4elab2g 2831 1 (𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wex 1468  wcel 1480  cop 3530  ran crn 4540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550
This theorem is referenced by:  elrng  4730  fvelrn  5551  fo2ndf  6124
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