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Theorem brxrn2 37847
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 37844 . . . . . . 7 (𝑅𝑆) ⊆ (V × (V × V))
21brel 5743 . . . . . 6 (𝐴(𝑅𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V)))
32simprd 495 . . . . 5 (𝐴(𝑅𝑆)𝐵𝐵 ∈ (V × V))
4 elvv 5752 . . . . 5 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4sylib 217 . . . 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 1947 . . 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 1844 . . 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 37846 . . . . . 6 ((𝐴𝑉𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1312el3v23 37697 . . . . 5 (𝐴𝑉 → (𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1413anbi2d 629 . . . 4 (𝐴𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥𝐴𝑆𝑦))))
15 3anass 1093 . . . 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 1920 . 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 205  wa 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  Vcvv 3471  cop 4635   class class class wbr 5148   × cxp 5676  cxrn 37647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fo 6554  df-fv 6556  df-1st 7993  df-2nd 7994  df-xrn 37843
This theorem is referenced by:  dfxrn2  37848  elecxrn  37858  inxpxrn  37867  br1cnvxrn2  37868
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