Theorem elrn2 4884

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 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elrn2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrn.1	. 2 ⊢ 𝐴 ∈ V
2		opeq2 3794	. . . 4 ⊢ (𝑦 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐴⟩)
3	2	eleq1d 2258	. . 3 ⊢ (𝑦 = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
4	3	exbidv 1836	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))
5		dfrn3 4831	. 2 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐵}
6	1, 4, 5	elab2 2900	1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2160  Vcvv 2752  ⟨cop 3610  ran crn 4642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4649  df-dm 4651  df-rn 4652

This theorem is referenced by:  elrn  4885  dmrnssfld  4905  rniun  5054  rnxpid  5078  ssrnres  5086  relssdmrn  5164