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Theorem rnxrn 38399
Description: Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.)
Assertion
Ref Expression
rnxrn ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}
Distinct variable groups:   𝑢,𝑅,𝑥,𝑦   𝑢,𝑆,𝑥,𝑦

Proof of Theorem rnxrn
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 3anass 1095 . . . . 5 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
213exbii 1850 . . . 4 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
3 exrot3 2165 . . . 4 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ ∃𝑥𝑦𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
4 19.42v 1953 . . . . 5 (∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
542exbii 1849 . . . 4 (∃𝑥𝑦𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
62, 3, 53bitri 297 . . 3 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
76abbii 2809 . 2 {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
8 dfrn6 38303 . . 3 ran (𝑅𝑆) = {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅}
9 n0 4353 . . . . 5 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆))
10 elec1cnvxrn2 38398 . . . . . . 7 (𝑢 ∈ V → (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
1110elv 3485 . . . . . 6 (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1211exbii 1848 . . . . 5 (∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
139, 12bitri 275 . . . 4 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1413abbii 2809 . . 3 {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅} = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
158, 14eqtri 2765 . 2 ran (𝑅𝑆) = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
16 df-opab 5206 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
177, 15, 163eqtr4i 2775 1 ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  Vcvv 3480  c0 4333  cop 4632   class class class wbr 5143  {copab 5205  ccnv 5684  ran crn 5686  [cec 8743  cxrn 38181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-ec 8747  df-xrn 38372
This theorem is referenced by:  rnxrnres  38400  dfcoss4  38416  dfssr2  38500
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