Theorem elrnres 39211

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 5872	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥 𝑥(𝑅 ↾ 𝐴)𝐵))
2		brres 5976	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥(𝑅 ↾ 𝐴)𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
3	2	exbidv 1953	. . 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 3087	. 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 1811   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102  ran crn 5651   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662

This theorem is used by:  elrnressn  39213