Theorem elrn2g 5872

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 4840	. . . 4 ⊢ (𝑦 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐴⟩)
2	1	eleq1d 2845	. . 3 ⊢ (𝑦 = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
3	2	exbidv 1953	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))
4		dfrn3 5871	. 2 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐵}
5	3, 4	elab2g 3633	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ⟨cop 4834  ran crn 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by:  elrng  5873  elrn2  5874  fvrnressn  7168  fo2ndf  8136