Theorem elrn2g 5837

Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Assertion

Ref	Expression
elrn2g	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 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.

Step	Hyp	Ref	Expression
1		opeq2 4828	. . . 4 ⊢ (𝑦 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐴⟩)
2	1	eleq1d 2813	. . 3 ⊢ (𝑦 = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
3	2	exbidv 1921	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))
4		dfrn3 5836	. 2 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐵}
5	3, 4	elab2g 3638	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ⟨cop 4585  ran crn 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-cnv 5631  df-dm 5633  df-rn 5634

This theorem is referenced by:  elrng  5838  elrn2  5839  fvrnressn  7099  fo2ndf  8061