Theorem elrng 5863

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

Assertion

Ref	Expression
elrng	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elrng

Step	Hyp	Ref	Expression
1		elrn2g 5862	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))
2		df-br 5098	. . 3 ⊢ (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
3	2	exbii 1867	. 2 ⊢ (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)
4	1, 3	bitr4di 291	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1798   ∈ wcel 2141  ⟨cop 4585   class class class wbr 5097  ran crn 5644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-cnv 5651  df-dm 5653  df-rn 5654

This theorem is referenced by:  elrn  5865  ssrelrn  5866  relelrnb  5919  cicsym  17828  elrnres  38738  eupre2  38953