Theorem elrnres 38314

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 5830	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ ran (𝑅 ↾ 𝐴) ↔ ∃𝑥 𝑥(𝑅 ↾ 𝐴)𝐵))
2		brres 5934	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥(𝑅 ↾ 𝐴)𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
3	2	exbidv 1922	. . 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 3057	. 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 1780   ∈ wcel 2111  ∃wrex 3056   class class class wbr 5089  ran crn 5615   ↾ cres 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626

This theorem is referenced by:  elrnressn  38316