Theorem elrnres 38253

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 5905	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥 𝑥(𝑅 ↾ 𝐴)𝐵))
2		brres 6007	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥(𝑅 ↾ 𝐴)𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
3	2	exbidv 1919	. . 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 3069	. 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 1776   ∈ wcel 2106  ∃wrex 3068   class class class wbr 5148  ran crn 5690   ↾ cres 5691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701

This theorem is referenced by:  elrnressn  38255