Theorem elrnres 38294

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 5876	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥 𝑥(𝑅 ↾ 𝐴)𝐵))
2		brres 5978	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥(𝑅 ↾ 𝐴)𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
3	2	exbidv 1921	. . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑥 𝑥(𝑅 ↾ 𝐴)𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
4	1, 3	bitrd 279	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
5		df-rex 3062	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵))
6	4, 5	bitr4di 289	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  ∃wrex 3061   class class class wbr 5124  ran crn 5660   ↾ cres 5661

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671

This theorem is referenced by:  elrnressn  38296