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Theorem brxrn2 36775
Description: A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020.)
Assertion
Ref Expression
brxrn2 (𝐴𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑉,𝑦

Proof of Theorem brxrn2
StepHypRef Expression
1 xrnss3v 36772 . . . . . . 7 (𝑅𝑆) ⊆ (V × (V × V))
21brel 5695 . . . . . 6 (𝐴(𝑅𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V)))
32simprd 496 . . . . 5 (𝐴(𝑅𝑆)𝐵𝐵 ∈ (V × V))
4 elvv 5704 . . . . 5 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4sylib 217 . . . 4 (𝐴(𝑅𝑆)𝐵 → ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
65pm4.71ri 561 . . 3 (𝐴(𝑅𝑆)𝐵 ↔ (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵))
7 19.41vv 1954 . . 3 (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵))
8 breq2 5107 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
98pm5.32i 575 . . . 4 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
1092exbii 1851 . . 3 (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
116, 7, 103bitr2i 298 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
12 brxrn 36774 . . . . . 6 ((𝐴𝑉𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1312el3v23 36621 . . . . 5 (𝐴𝑉 → (𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1413anbi2d 629 . . . 4 (𝐴𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥𝐴𝑆𝑦))))
15 3anass 1095 . . . 4 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1614, 15bitr4di 288 . . 3 (𝐴𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
17162exbidv 1927 . 2 (𝐴𝑉 → (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
1811, 17bitrid 282 1 (𝐴𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  Vcvv 3443  cop 4590   class class class wbr 5103   × cxp 5629  cxrn 36571
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7913  df-2nd 7914  df-xrn 36771
This theorem is referenced by:  dfxrn2  36776  elecxrn  36786  inxpxrn  36795  br1cnvxrn2  36796
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