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Theorem brxrn2 34566
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 34563 . . . . . . 7 (𝑅𝑆) ⊆ (V × (V × V))
21brel 5336 . . . . . 6 (𝐴(𝑅𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V)))
32simprd 489 . . . . 5 (𝐴(𝑅𝑆)𝐵𝐵 ∈ (V × V))
4 elvv 5345 . . . . 5 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4sylib 209 . . . 4 (𝐴(𝑅𝑆)𝐵 → ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
65pm4.71ri 556 . . 3 (𝐴(𝑅𝑆)𝐵 ↔ (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵))
7 19.41vv 2045 . . 3 (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵))
8 breq2 4813 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
98pm5.32i 570 . . . 4 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
1092exbii 1944 . . 3 (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
116, 7, 103bitr2i 290 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
12 brxrn 34565 . . . . . 6 ((𝐴𝑉𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1312el3v23 34434 . . . . 5 (𝐴𝑉 → (𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1413anbi2d 622 . . . 4 (𝐴𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥𝐴𝑆𝑦))))
15 3anass 1116 . . . 4 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1614, 15syl6bbr 280 . . 3 (𝐴𝑉 → ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
17162exbidv 2019 . 2 (𝐴𝑉 → (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
1811, 17syl5bb 274 1 (𝐴𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  cop 4340   class class class wbr 4809   × cxp 5275  cxrn 34403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-1st 7366  df-2nd 7367  df-xrn 34562
This theorem is referenced by:  dfxrn2  34567  elecxrn  34577  inxpxrn  34582  br1cnvxrn2  34583
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