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Theorem rnxrnidres 37209
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⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}
Distinct variable groups:   𝑢,𝐴,𝑥,𝑦   𝑢,𝑅,𝑥,𝑦

Proof of Theorem rnxrnidres
StepHypRef Expression
1 rnxrnres 37207 . 2 ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 I 𝑦)}
2 ideqg 5849 . . . . . 6 (𝑦 ∈ V → (𝑢 I 𝑦𝑢 = 𝑦))
32elv 3481 . . . . 5 (𝑢 I 𝑦𝑢 = 𝑦)
43anbi1ci 627 . . . 4 ((𝑢𝑅𝑥𝑢 I 𝑦) ↔ (𝑢 = 𝑦𝑢𝑅𝑥))
54rexbii 3095 . . 3 (∃𝑢𝐴 (𝑢𝑅𝑥𝑢 I 𝑦) ↔ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥))
65opabbii 5214 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 I 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}
71, 6eqtri 2761 1 ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wrex 3071  Vcvv 3475   class class class wbr 5147  {copab 5209   I cid 5572  ran crn 5676  cres 5677  cxrn 36980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7970  df-2nd 7971  df-ec 8701  df-xrn 37179
This theorem is referenced by: (None)
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