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Theorem brxrn2 38357
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 38354 . . . . . . 7 (𝑅𝑆) ⊆ (V × (V × V))
21brel 5754 . . . . . 6 (𝐴(𝑅𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V)))
32simprd 495 . . . . 5 (𝐴(𝑅𝑆)𝐵𝐵 ∈ (V × V))
4 elvv 5763 . . . . 5 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4sylib 218 . . . 4 (𝐴(𝑅𝑆)𝐵 → ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
65pm4.71ri 560 . . 3 (𝐴(𝑅𝑆)𝐵 ↔ (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵))
7 19.41vv 1948 . . 3 (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵))
8 breq2 5152 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
98pm5.32i 574 . . . 4 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
1092exbii 1846 . . 3 (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
116, 7, 103bitr2i 299 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
12 brxrn 38356 . . . . . 6 ((𝐴𝑉𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1312el3v23 38209 . . . . 5 (𝐴𝑉 → (𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1413anbi2d 630 . . . 4 (𝐴𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥𝐴𝑆𝑦))))
15 3anass 1094 . . . 4 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1614, 15bitr4di 289 . . 3 (𝐴𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
17162exbidv 1922 . 2 (𝐴𝑉 → (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
1811, 17bitrid 283 1 (𝐴𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148   × cxp 5687  cxrn 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-xrn 38353
This theorem is referenced by:  dfxrn2  38358  elecxrn  38368  inxpxrn  38377  br1cnvxrn2  38378
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