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Theorem rnxrn 37073
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 1852 . . . 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 1957 . . . . 5 (∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
542exbii 1851 . . . 4 (∃𝑥𝑦𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
62, 3, 53bitri 296 . . 3 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
76abbii 2801 . 2 {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
8 dfrn6 36976 . . 3 ran (𝑅𝑆) = {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅}
9 n0 4342 . . . . 5 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆))
10 elec1cnvxrn2 37072 . . . . . . 7 (𝑢 ∈ V → (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
1110elv 3479 . . . . . 6 (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1211exbii 1850 . . . . 5 (∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
139, 12bitri 274 . . . 4 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1413abbii 2801 . . 3 {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅} = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
158, 14eqtri 2759 . 2 ran (𝑅𝑆) = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
16 df-opab 5204 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
177, 15, 163eqtr4i 2769 1 ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2708  wne 2939  Vcvv 3473  c0 4318  cop 4628   class class class wbr 5141  {copab 5203  ccnv 5668  ran crn 5670  [cec 8684  cxrn 36847
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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fo 6538  df-fv 6540  df-1st 7957  df-2nd 7958  df-ec 8688  df-xrn 37046
This theorem is referenced by:  rnxrnres  37074  dfcoss4  37090  dfssr2  37174
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