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Theorem rnxrnres 36262
Description: Range of a range Cartesian product with a restricted relation. (Contributed by Peter Mazsa, 5-Dec-2021.)
Assertion
Ref Expression
rnxrnres ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
Distinct variable groups:   𝑢,𝐴,𝑥,𝑦   𝑢,𝑅,𝑥,𝑦   𝑢,𝑆,𝑥,𝑦

Proof of Theorem rnxrnres
StepHypRef Expression
1 rnxrn 36261 . 2 ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦)}
2 brres 5858 . . . . . . . 8 (𝑦 ∈ V → (𝑢(𝑆𝐴)𝑦 ↔ (𝑢𝐴𝑢𝑆𝑦)))
32elv 3414 . . . . . . 7 (𝑢(𝑆𝐴)𝑦 ↔ (𝑢𝐴𝑢𝑆𝑦))
43anbi2i 626 . . . . . 6 ((𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ (𝑢𝑅𝑥 ∧ (𝑢𝐴𝑢𝑆𝑦)))
5 an12 645 . . . . . 6 ((𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ (𝑢𝑅𝑥 ∧ (𝑢𝐴𝑢𝑆𝑦)))
64, 5bitr4i 281 . . . . 5 ((𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ (𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
76exbii 1855 . . . 4 (∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
8 df-rex 3067 . . . 4 (∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
97, 8bitr4i 281 . . 3 (∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦))
109opabbii 5120 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
111, 10eqtri 2765 1 ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2110  wrex 3062  Vcvv 3408   class class class wbr 5053  {copab 5115  ran crn 5552  cres 5553  cxrn 36069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388  df-1st 7761  df-2nd 7762  df-ec 8393  df-xrn 36238
This theorem is referenced by:  rnxrncnvepres  36263  rnxrnidres  36264
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