Theorem rnxrnidres 38548

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 38546	. 2 ⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦)}
2		ideqg 5798	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑢 I 𝑦 ↔ 𝑢 = 𝑦))
3	2	elv 3443	. . . . 5 ⊢ (𝑢 I 𝑦 ↔ 𝑢 = 𝑦)
4	3	anbi1ci 626	. . . 4 ⊢ ((𝑢𝑅𝑥 ∧ 𝑢 I 𝑦) ↔ (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥))
5	4	rexbii 3081	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦) ↔ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥))
6	5	opabbii 5163	. 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 1541  ∃wrex 3058  Vcvv 3438   class class class wbr 5096  {copab 5158   I cid 5516  ran crn 5623   ↾ cres 5624   ⋉ cxrn 38314

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-1st 7931  df-2nd 7932  df-ec 8635  df-xrn 38504

This theorem is referenced by: (None)