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Theorem elrn2 4604
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 3578 . . . 4 (𝑦 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐴⟩)
32eleq1d 2122 . . 3 (𝑦 = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
43exbidv 1722 . 2 (𝑦 = 𝐴 → (∃𝑥𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
5 dfrn3 4552 . 2 ran 𝐵 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐵}
61, 4, 5elab2 2713 1 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  cop 3406  ran crn 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-cnv 4381  df-dm 4383  df-rn 4384
This theorem is referenced by:  elrn  4605  dmrnssfld  4623  rniun  4762  rnxpid  4783  ssrnres  4791  relssdmrn  4869
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