Theorem rnxrnidres 34706

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 34704	. 2 ⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦)}
2		ideqg 5505	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑢 I 𝑦 ↔ 𝑢 = 𝑦))
3	2	elv 3417	. . . . 5 ⊢ (𝑢 I 𝑦 ↔ 𝑢 = 𝑦)
4	3	anbi1ci 621	. . . 4 ⊢ ((𝑢𝑅𝑥 ∧ 𝑢 I 𝑦) ↔ (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥))
5	4	rexbii 3250	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦) ↔ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥))
6	5	opabbii 4939	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢 I 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)}
7	1, 6	eqtri 2848	1 ⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)}

Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1658  ∃wrex 3117  Vcvv 3413   class class class wbr 4872  {copab 4934   I cid 5248  ran crn 5342   ↾ cres 5343   ⋉ cxrn 34522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-fo 6128  df-fv 6130  df-1st 7427  df-2nd 7428  df-ec 8010  df-xrn 34680

This theorem is referenced by: (None)