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Theorem rnxrnres 38401
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 38400 . 2 ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦)}
2 brres 6003 . . . . . . . 8 (𝑦 ∈ V → (𝑢(𝑆𝐴)𝑦 ↔ (𝑢𝐴𝑢𝑆𝑦)))
32elv 3484 . . . . . . 7 (𝑢(𝑆𝐴)𝑦 ↔ (𝑢𝐴𝑢𝑆𝑦))
43anbi2i 623 . . . . . 6 ((𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ (𝑢𝑅𝑥 ∧ (𝑢𝐴𝑢𝑆𝑦)))
5 an12 645 . . . . . 6 ((𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ (𝑢𝑅𝑥 ∧ (𝑢𝐴𝑢𝑆𝑦)))
64, 5bitr4i 278 . . . . 5 ((𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ (𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
76exbii 1847 . . . 4 (∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
8 df-rex 3070 . . . 4 (∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
97, 8bitr4i 278 . . 3 (∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦))
109opabbii 5209 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
111, 10eqtri 2764 1 ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wex 1778  wcel 2107  wrex 3069  Vcvv 3479   class class class wbr 5142  {copab 5204  ran crn 5685  cres 5686  cxrn 38182
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 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-1st 8015  df-2nd 8016  df-ec 8748  df-xrn 38373
This theorem is referenced by:  rnxrncnvepres  38402  rnxrnidres  38403
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