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Theorem rnxrnres 35527
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 35526 . 2 ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦)}
2 brres 5853 . . . . . . . 8 (𝑦 ∈ V → (𝑢(𝑆𝐴)𝑦 ↔ (𝑢𝐴𝑢𝑆𝑦)))
32elv 3497 . . . . . . 7 (𝑢(𝑆𝐴)𝑦 ↔ (𝑢𝐴𝑢𝑆𝑦))
43anbi2i 622 . . . . . 6 ((𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ (𝑢𝑅𝑥 ∧ (𝑢𝐴𝑢𝑆𝑦)))
5 an12 641 . . . . . 6 ((𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ (𝑢𝑅𝑥 ∧ (𝑢𝐴𝑢𝑆𝑦)))
64, 5bitr4i 279 . . . . 5 ((𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ (𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
76exbii 1839 . . . 4 (∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
8 df-rex 3141 . . . 4 (∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
97, 8bitr4i 279 . . 3 (∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦))
109opabbii 5124 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
111, 10eqtri 2841 1 ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  Vcvv 3492   class class class wbr 5057  {copab 5119  ran crn 5549  cres 5550  cxrn 35333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-2nd 7679  df-ec 8280  df-xrn 35503
This theorem is referenced by:  rnxrncnvepres  35528  rnxrnidres  35529
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