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Theorem rnxrnres 35826
 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 35825 . 2 ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦)}
2 brres 5826 . . . . . . . 8 (𝑦 ∈ V → (𝑢(𝑆𝐴)𝑦 ↔ (𝑢𝐴𝑢𝑆𝑦)))
32elv 3446 . . . . . . 7 (𝑢(𝑆𝐴)𝑦 ↔ (𝑢𝐴𝑢𝑆𝑦))
43anbi2i 625 . . . . . 6 ((𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ (𝑢𝑅𝑥 ∧ (𝑢𝐴𝑢𝑆𝑦)))
5 an12 644 . . . . . 6 ((𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ (𝑢𝑅𝑥 ∧ (𝑢𝐴𝑢𝑆𝑦)))
64, 5bitr4i 281 . . . . 5 ((𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ (𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
76exbii 1849 . . . 4 (∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
8 df-rex 3112 . . . 4 (∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
97, 8bitr4i 281 . . 3 (∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦) ↔ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦))
109opabbii 5098 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢(𝑆𝐴)𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
111, 10eqtri 2821 1 ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441   class class class wbr 5031  {copab 5093  ran crn 5521   ↾ cres 5522   ⋉ cxrn 35631 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fo 6331  df-fv 6333  df-1st 7674  df-2nd 7675  df-ec 8277  df-xrn 35802 This theorem is referenced by:  rnxrncnvepres  35827  rnxrnidres  35828
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