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Theorem rnxrn 34475
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 1081 . . . . 5 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
213exbii 1921 . . . 4 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
3 exrot3 2190 . . . 4 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ ∃𝑥𝑦𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)))
4 19.42v 2026 . . . . 5 (∃𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
542exbii 1920 . . . 4 (∃𝑥𝑦𝑢(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥𝑢𝑆𝑦)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
62, 3, 53bitri 286 . . 3 (∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)))
76abbii 2873 . 2 {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
8 dfrn6 34392 . . 3 ran (𝑅𝑆) = {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅}
9 n0 4070 . . . . 5 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆))
10 elec1cnvxrn2 34474 . . . . . . 7 (𝑢 ∈ V → (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)))
1110elv 34305 . . . . . 6 (𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1211exbii 1919 . . . . 5 (∃𝑢 𝑢 ∈ [𝑤](𝑅𝑆) ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
139, 12bitri 264 . . . 4 ([𝑤](𝑅𝑆) ≠ ∅ ↔ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦))
1413abbii 2873 . . 3 {𝑤 ∣ [𝑤](𝑅𝑆) ≠ ∅} = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
158, 14eqtri 2778 . 2 ran (𝑅𝑆) = {𝑤 ∣ ∃𝑢𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥𝑢𝑆𝑦)}
16 df-opab 4861 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦))}
177, 15, 163eqtr4i 2788 1 ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1628  wex 1849  wcel 2135  {cab 2742  wne 2928  Vcvv 3336  c0 4054  cop 4323   class class class wbr 4800  {copab 4860  ccnv 5261  ran crn 5263  [cec 7905  cxrn 34291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-fo 6051  df-fv 6053  df-1st 7329  df-2nd 7330  df-ec 7909  df-xrn 34452
This theorem is referenced by:  rnxrnres  34476  dfcoss4  34492  dfssr2  34568
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