Theorem elrnres 39013

Description: Element of the range of a restriction. (Contributed by Peter Mazsa, 26-Dec-2018.)

Assertion

Ref	Expression
elrnres	⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉

Proof of Theorem elrnres

Step	Hyp	Ref	Expression
1		elrng 5879	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥 𝑥(𝑅 ↾ 𝐴)𝐵))
2		brres 5983	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥(𝑅 ↾ 𝐴)𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
3	2	exbidv 1954	. . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑥 𝑥(𝑅 ↾ 𝐴)𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
4	1, 3	bitrd 282	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
5		df-rex 3089	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵))
6	4, 5	bitr4di 292	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088   class class class wbr 5107  ran crn 5660   ↾ cres 5661

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 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671

This theorem is used by:  elrnressn  39015