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Theorem elrn2 5797
Description: Membership in a range. (Contributed by NM, 10-Jul-1994.)
Hypothesis
Ref Expression
elrn.1 𝐴 ∈ V
Assertion
Ref Expression
elrn2 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elrn2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elrn.1 . 2 𝐴 ∈ V
2 opeq2 4780 . . . 4 (𝑦 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐴⟩)
32eleq1d 2895 . . 3 (𝑦 = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
43exbidv 1922 . 2 (𝑦 = 𝐴 → (∃𝑥𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
5 dfrn3 5736 . 2 ran 𝐵 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐵}
61, 4, 5elab2 3650 1 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 208   = wceq 1537  wex 1780  wcel 2114  Vcvv 3473  cop 4549  ran crn 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-cnv 5539  df-dm 5541  df-rn 5542
This theorem is referenced by:  elrn  5798  dmrnssfld  5817  rniun  5982  ssrnres  6011  relssdmrn  6097  fvelrn  6820  tz7.48-1  8057  prsrn  31166  dfrn5  33025  funressndmafv2rn  43570
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