Theorem elrng 5883

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 5882	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))
2		df-br 5112	. . 3 ⊢ (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
3	2	exbii 1881	. 2 ⊢ (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵)
4	1, 3	bitr4di 292	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812   ∈ wcel 2146  ⟨cop 4597   class class class wbr 5111  ran crn 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-cnv 5671  df-dm 5673  df-rn 5674

This theorem is used by:  elrn  5885  ssrelrn  5886  relelrnb  5939  cicsym  17878  elrnres  38960  eupre2  39175