Theorem rnxrnidres 38361

Description: Range of a range Cartesian product with a restriction of the identity relation. (Contributed by Peter Mazsa, 6-Dec-2021.)

Assertion

Ref	Expression
rnxrnidres	⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)}

Distinct variable groups:   𝑢,𝐴,𝑥,𝑦   𝑢,𝑅,𝑥,𝑦

Proof of Theorem rnxrnidres

Step	Hyp	Ref	Expression
1		rnxrnres 38359	. 2 ⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦)}
2		ideqg 5842	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑢 I 𝑦 ↔ 𝑢 = 𝑦))
3	2	elv 3468	. . . . 5 ⊢ (𝑢 I 𝑦 ↔ 𝑢 = 𝑦)
4	3	anbi1ci 626	. . . 4 ⊢ ((𝑢𝑅𝑥 ∧ 𝑢 I 𝑦) ↔ (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥))
5	4	rexbii 3082	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦) ↔ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥))
6	5	opabbii 5190	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)}
7	1, 6	eqtri 2757	1 ⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wrex 3059  Vcvv 3463   class class class wbr 5123  {copab 5185   I cid 5557  ran crn 5666   ↾ cres 5667   ⋉ cxrn 38140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  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-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fo 6547  df-fv 6549  df-1st 7996  df-2nd 7997  df-ec 8729  df-xrn 38331

This theorem is referenced by: (None)